บทที่-3-เรื่อง-จำนวนจริง-คณิตศาสตร์เพิ่มเติม-ม.4-เทอม-1

เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพ่มิ เติม ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 51

11. จงหาจำนวนจรงิ , และ ทที่ ำใหส2 มการในแตลB ะข2อตBอไปน้เี ป7นจริง
1) (- '( -& )'(- ),%) 7 -8,%
= - ' & +



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


2) &--!,!,!-&- ' #
= 3 + 7 + 8--",,&9
-

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


3) -!!-,"# -'" - ', ((-
= 7 + 8 − -,9 (
- - ' %


………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

7-,8 " - ') !
" ' (-,9 - ' &
4) - = +

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พิ่มเติม ม.4 เทอม 1 เรือ่ ง จำนวนจริง 52

12. จงหาเซตคำตอบของสมการตBอไปนี้
1) --((-- ' ' &&))((-- ,'! !)) = 0
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


2) (-,-!(-)(,-& ') %) = (-,!)#(- ' %)

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

! ,&-",- ' %
-" ' %
3) - = 0

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


4) - '% % %
+ -,% = 1

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพ่มิ เติม ม.4 เทอม 1 เร่ือง จำนวนจริง 53

5) - " % + % = -,& %
' % - ' !

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

% % %
' !- ' + -" ' (-,# -",- ' !
6) - " + + = 0

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

7) -,- # % !-
-,! -","-,+
+ + = 0

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณิตศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรือ่ ง จำนวนจริง 54

8) !!--"",,(-- ' ' & ) = !-%,&



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


9) -!"- ', %% ! -"(,-
− - ' ! =

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

% % - '! !
" ' &-,! -" ' %
10) - + =

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พิม่ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจรงิ 55

13. จงหาเซตคำตอบของอสมการตอB ไปน ้ี

1) 2( + 1) < + 2



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


2) 4 + 7 > 2( + 1)




………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


3) 4 − (3 − ) < 3 − (3 − 2 )




………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


4) 2 ! − − 6 ≥ 0




………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


5) ! ≥ 2 − 3



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรอ่ื ง จำนวนจรงิ 56

6) ( + 1) ≤ 20



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................



7) ( − 1)( − 4) > ( − 2)( − 3)



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


8) & + 4 > 3 !



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


9) ( − 1)( − 2)( − 3) < ( − 1)( − 2)



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


10) ( − 2)( − 3)!( − 4) ≤ 0



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เรอื่ ง จำนวนจริง 57

11) ( − 2)( − 3)!( − 4) ≥ 0



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


12) ( + 1)(4 − )( − 6)! ≥ 0



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................



14. จงหาเซตคำตอบของอสมการตBอไปน ้ี

1) -% > -,%%



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


2) - '& % ≤ 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพมิ่ เตมิ ม.4 เทอม 1 เร่อื ง จำนวนจริง 58

3) !-- ' ' % # ≥ 2



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


4) --,,%! < 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


5) -,%% ≥ -,%#



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


6) -,%! ≤ !-% ' &



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณิตศาสตร2เพม่ิ เติม ม.4 เทอม 1 เรือ่ ง จำนวนจริง 59

7) + # ≤ 4
-

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


8) --", '% & < + 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


9) !--"" ' ' ! "-- ', &% ≤ 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพ่มิ เติม ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 60

10) (- ' %! )'( -- ' () ≤ 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

- %!
",%
11) - ≤

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

-
",!
12) - ≥ 3

………………………………………………………………............................…….………………………………………………………………....................................

………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณติ ศาสตร2เพมิ่ เตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 61

13) -%"% ' ' - ( '- ! ≤ 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


14) ((-- '' !%))((-- '' &!))((-- '' &#)) ≤ 0



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


15) 4-",&-- "', %!/-5 4'- %"(,- ' "5 ≤ 0



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณติ ศาสตรเ2 พิ่มเติม ม.4 เทอม 1 เรอื่ ง จำนวนจริง 62

15. โรงงานผลติ กลBองดนิ สอแหงB หนึ่งมคี Bาใชจ2 Bายในการผลติ กลBองดนิ สอกลBองละ 26 บาท และมีคBาใช2จBายอ่นื ๆ
อกี สปั ดาหRละ 30,000 บาท ถ2าโรงงานขายกลอB งดินสอกลอB งละ 30 บาท
จงหาวBา ในหนง่ึ สัปดาหRโรงงานจะตอ2 งผลิตกลBองดนิ สออยาB งนอ2 ยท่ีสดุ กี่กลBองจึงจะไมBขาดทุน

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................


16. รายไดข2 องบรษิ ัทแหงB หนง่ึ สอดคล2องกบั สมการ ( ) = 30 ! − 35940 − 72000
เมอ่ื แทนจำนวนสินคา2 ทีบ่ ริษทั ผลิตและจำหนาB ย (ช้ิน)
( ) แทนรายได2จากการขายสนิ ค2า ชน้ิ (บาท)
จงหาวBา บรษิ ทั จะต2องผลิตและจำหนาB ยสินค2าอยBางน2อยทสี่ ดุ ก่ีชิ้นจึงจะไมBขาดทุน

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................


17. ถ2าฐานของรปู สามเหล่ยี มรปู หนง่ึ สั้นกวาB สBวนสูง 5 เซนติเมตร พน้ื ทขี่ องรูปสามเหล่ียมนี้จะมคี าB อยรูB ะหวBาง
42 และ 52 ตารางเซนติเมตร ความยาวของฐานควรอยใBู นชBวงใด

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................

เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พ่มิ เติม ม.4 เทอม 1 เรื่อง จำนวนจรงิ 63

18. จำนวนคสี่ ามจำนวนที่เรียงตดิ กนั คูณกันแลว2 ไมมB ากกวBา 315 จงหาผลคณู ที่มากท่ีสดุ ท่เี ปน7 ไปได 2
ของท้งั สามจำนวน

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................


19. ชBางตัดเสอ้ื ซือ้ ผ2ามาท้งั สิ้น 600 บาท ตดั เก็บไว 2 5 เมตร ทีเ่ หลือขายไปในราคาสงู กวBาต2นทุน
เมตรละ 10 บาท ได2กำไร 80 บาท ชBางตัดเสอื้ ซ้อื ผา2 มาราคาเมตรละเทาB ใด

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................

เอกสารประกอบการสอนวชิ าคณิตศาสตร2เพิ่มเตมิ ม.4 เทอม 1 เรอ่ื ง จำนวนจริง 64

20. จงหาเซตคำตอบของสมการตอB ไปน้ี

1) | − 2| = 2



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................



2) |2 − 1| = + 4



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................



3) |3 − 1| = | − 5|



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พิม่ เตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 65

4) | ! − 3 + 1| = − 2



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


5) | ! + 2 − 1| = + 5



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


6) | ! − 3 − 4| = |2 + 2|



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพิม่ เตมิ ม.4 เทอม 1 เร่ือง จำนวนจรงิ 66

7) | | + | − 3| = 2



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


8) 4| | = | − 2| + 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

9) | − 1| + | − 2| = 3



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตร2เพมิ่ เติม ม.4 เทอม 1 เรือ่ ง จำนวนจริง 67

10) | | − | − 2| = − 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................



21. จงหาเซตคำตอบของอสมการตอB ไปน ี้

1) |2 − 4| > + 1



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

2) | − 4| ≤ 2 + 1




………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

3) |2 − 3| < 3 − 7



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณิตศาสตรเ2 พิม่ เตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 68

4) | ! − 4| ≤ | ! − 2 |



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................


5) |2 ! − 5 − 1| ≤ | ! − + 4|



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

6) 2 _-- '' (%_ ≤ _-- '' (#_



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวิชาคณติ ศาสตรเ2 พม่ิ เตมิ ม.4 เทอม 1 เรอื่ ง จำนวนจรงิ 69

7) |-|% ' ! ≥ |-|!,%



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

8) |--,|,!" + 1 < -| '-| &



………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

9) _- '- &_ ≥ --,,(!
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................
………………………………………………………………............................…….………………………………………………………………....................................

เอกสารประกอบการสอนวชิ าคณิตศาสตรเ2 พ่ิมเตมิ ม.4 เทอม 1 เรือ่ ง จำนวนจริง 70

22. สมชายเดินทางออกจากทท่ี ำงาน ซึง่ ตัง้ อยทBู ี่หลักกโิ ลเมตรที่ 5 ของถนนสายหนึง่ เพ่อื ไปซอื้ ของทรี่ 2านค2า
ซง่ึ ตั้งอยูบB นถนนสายเดียวกนั จากน้นั จึงเดนิ ทางกลับบ2านของตนเองซ่ึงตง้ั อยทBู ่หี ลกั กิโลเมตรที่ 7
ของถนนสายดงั กลาB ว พบวBาไดเ2 ดนิ ทางไปเปน7 ระยะทางทง้ั สนิ้ 4 กิโลเมตรจงหาตำแหนงB ท่ีตั้ง (ทเี่ ปน7 ไปได)2
ของร2านคา2 ดังกลาB ว

………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................
………………………………………………………………............................…….……………………………………………………………….......................................................