จำนวนจริง

44 ⌫ ⌫  ⌦
 ⌫     ⌫  

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦¼o
4.1 ššªœ„µ¦Â¥„˜ª´ ž¦³„° ¨³„µ¦Â„o­¤„µ¦˜ª´ ž¦Á—¥¸ ª ×¥‡¦Â¼ ‹„Á°„­µ¦Âœ³Âœª

šµŠš¸É 6.1 Ä®œo „´ Á¦¥¸ œ‹´ ‡¼n¨ªo ªn ¥„´œ‡—· čoÁª¨µ 10 œµš¸
4.2 ‡¦Á¼ Œ¨¥‡µÎ ˜°œ„¦³—µœ—µÎ ¨³°›· µ¥Á¡·¤É Á˜¤· …°o šÉœ¸ ´„Á¦¥¸ œÅ¤Án …µo ċ Áœn ‹ŠÂ„o­¤„µ¦

2x3-3x2-5x+6 = 0 œ´„Á¦¸¥œ­nªœÄ®‹n ³®µ‡µÎ ˜°Å¤Ån —o Á¡¦µ³œ´„Á¦¥¸ œÅ¤­n µ¤µ¦™Â¥„˜ª´ ž¦³„°Å—o
ץčo‡ªµ¤¦o¼ ´œÊ ¤›´ ¥¤«¹„¬µ˜°œ˜œo

4.3 ‡¦¼°„œ´„Á¦¥¸ œªµn Á¦µ‹³Äoš§¬‘¸šÁ«¬Á®¨º°š¸ÅÉ —oÁ¦¥¸ œ¤µÂ¨oª ¨³š§¬‘¸ššÁ¸É „ɸ¥ª„´
˜ª´ ž¦³„° Á¡ºÉ°œµÎ ¤µÄoĜ„µ¦Â¥„˜ª´ ž¦³„°­Îµ®¦´Ã‹š¥¨r „´ ¬–³Á—¥¸ ª„´ …°o 2.5 ¨ªo ‡¦¼Â¨³œ´„Á¦¸¥œ
ªn ¥„œ´ ššªœš§¬‘¸šÁ«¬Á®¨º° ×¥‡¦™¼ µ¤Ä®oœ´„Á¦¥¸ œªn ¥„œ´ ˜°

4.4 ‡¦„¼ 宜—Ëš¥rÄ®œo „´ Á¦¥¸ œ®µÁ«¬‹µ„„µ¦®µ¦¡®œ» µ¤ ץčšo §¬‘¸ šÁ«¬Á®¨°º Áœn
x3-5x2+2x+8 ®µ¦—ªo ¥ x-2 ŽŠ¹É œ„´ Á¦¥¸ œ‡ª¦®µÅ—ªo µn Á«¬ = 0

4.5 nŠ„¨»n¤œ´„Á¦¸¥œ„¨¤»n ¨³ 4-5 ‡œ ‡¦Â¼ ‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 6.2 Ä®œo „´ Á¦¸¥œš»„‡œªn ¥
„´œ«„¹ ¬µš§¬‘¸š˜ª´ ž¦³„° ¨³˜°‡µÎ ™µ¤ čÁo ª¨µ 15 œµš¸ ¨oª‡¦¼Äªo ·›„¸ µ¦™µ¤˜°Á¡É°º ÁŒ¨¥
‡Îµ˜°Â¨³°›· µ¥Ä®šo „» ‡œÁ…µo ċ˜¦Š„´œ ™µo „¨»n¤Ä—¤¸ž{ ®µ‹³°›·µ¥Á¡É·¤Á˜·¤

4.6 ‡¦¼„µÎ ®œ—Ëš¥rÁ„¥É¸ ª„´ „µ¦Â¥„˜ª´ ž¦³„°°„¸ 1 …o° Ä®Âo ˜n¨³„¨¤n» ªn ¥„œ´ ‡·—¨³­nŠ
˜ª´ šœ°°„¤µšµÎ œ„¦³—µœ—µÎ ¨oªÄ®Áo ¡°Éº œÇ Ĝ®°o Šªn ¥„´œ˜¦ª‹‡ªµ¤™„¼ ˜°o Š

4.7 Ä®œo ´„Á¦¥¸ œšÎµÂ „f ®´— 2.3 …°o 3 Ĝ®œŠ´ ­º°Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡¤·É Á˜·¤ ‡–·˜«µ­˜¦r
Á¨¤n 1 …°Š ­­ªš.

4.8 ‡¦Á¼ Œ¨¥Â „f ®—´ 2.3 ÁŒ¡µ³…o°šÉ¸œ´„Á¦¥¸ œ­Š­´¥®¦º°šÉ¸œ„´ Á¦¥¸ œ­ªn œÄ®šn εŸ—· ‡¦¼ ¸ÄÊ ®o
œ´„Á¦¸¥œÁ®œÈ ªµn ­Îµ®¦´¡®œ» µ¤ anxn+ an-1xn-1+ an-2xn-2+…+ a1x + a0 Á¤º°É anz1 ‹³¤¸š§¬‘¸š¤µnª¥
Ĝ„µ¦Â¥„˜ª´ ž¦³„°°¸„ 1 š§¬‘¸ š

4.9 ‡¦Â¼ ‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 6.3 Ä®šo »„‡œªn ¥„´œ«„¹ ¬µ (č„o ¨n»¤Äœ‡µšÉ¸ 1) š§¬‘¸š
˜ª´ ž¦³„°‹µÎ œªœ˜¦¦„¥³Â¨³˜ª´ °¥µn ŠÄoÁª¨µ 15 œµš¸ ‡¦¼°›· µ¥š§¬‘¸šÁ¡É¤· Á˜¤· Á¡º°É Áž}œ„µ¦Áœoœ
¨³­Îµ®¦´ „¨n»¤šÅ¸É ¤n‡°n ¥Á…µo ċ ®¨Š´ ‹µ„Á…µo ċš§¬‘¸š—¸Â¨ªo Ä®ošÎµÃ‹š¥Är œÁ°„­µ¦Âœ³ÂœªšµŠ

4.10 ­»n¤„¨»¤n Ä®˜o ª´ šœ°°„¤µÂ­—Šª›· ¸šµÎ œ„¦³—µœ ×¥‡¦Â¼ ¨³Á¡É°º œÇ nª¥„´œ˜¦ª‹­°
‡ªµ¤™„¼ ˜o°Š

4.11 Ä®œo „´ Á¦¥¸ œšÎµÂ „f ®´— 2.3 …o° 4 Ĝ®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡É¤· Á˜¤· ‡–˜· «µ­˜¦r
Á¨¤n 1 …°Š ­­ªš. ™µo ŤÁn ­¦È‹Ä®šo µÎ Ážœ} „µ¦µo œ

4.12 Áž—d ð„µ­Ä®œo ´„Á¦¸¥œŽ„´ ™µ¤…°o ­Š­¥´ Á„¸¥É ª„´„µ¦Â„­o ¤„µ¦˜´ªÂž¦Á—¥¸ ªÁ¡¤·É Á˜·¤ ¨³œ´—
®¤µ¥™¹Š„µ¦š—­°Äœ‡µ˜°n Ş

4.13 ÁŒ¨¥Â „f ®´—šÉ¸ 2.3 ÁŒ¡µ³…°o šœ¸É „´ Á¦¸¥œ­Š­´¥ ®¦°º œ„´ Á¦¥¸ œ­nªœÄ®n¤ž¸ {®µ ¨ªo š—­°
Á„È ‡³ÂœœÄÁo ª¨µ 30 œµš¸

⌦ 45
⌦

5. ®¨nŠ„µ¦Á¦¸¥œ¦¼o
5.1 ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Á¼o ¡¤É· Á˜¤· ‡–˜· «µ­˜¦r Á¨n¤ 1 …°Š ­­ªš.
5.2 Á°„­µ¦Âœ³ÂœªšµŠ 6.1 - 6.3

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª´—Ÿ¨ „µ¦ž¦³Á¤·œŸ¨
1. ­Š´ Á„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™¼„˜o°ŠÁž}œ­nªœ¤µ„
2. ­´ŠÁ„˜‹µ„‡ªµ¤­œÄ‹ 2. œ´„Á¦¥¸ œ­œÄ‹Â¨³˜´ÊŠÄ‹Á¦¥¸ œ
3. šµÎ Á°„­µ¦Âœ³ÂœªšµŠš¸É 6.1 , 6.2 , 6.3 3. œ„´ Á¦¸¥œšµÎ ŗo™„¼ ˜°o Šž¦³¤µ– 80 %
4. šµÎ ˚¥Âr  f„®—´ 2.3 Ĝ®œŠ´ ­º°­µ¦³ 4. œ„´ Á¦¥¸ œšµÎ ŗ™o „¼ ˜°o Šž¦³¤µ– 85 %

„µ¦Á¦¥¸ œ¦Á¼o ¡É¤· Á˜¤· ‡–·˜«µ­˜¦r Á¨n¤ 1

7. œ´ š„¹ ®¨Š´ ­°œ
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8. „‹· „¦¦¤Á­œ°Âœ³
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46 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 6.1

‡Îµ¸Ê‹Š Ä®œo „´ Á¦¥¸ œÄ‡o ªµ¤¦šo¼ ÉÁ¸ ‡¥Á¦¥¸ œÄœœ´Ê ¤›´ ¥¤«„¹ ¬µ˜°œ˜œo ˜°‡µÎ ™µ¤˜°n Şœ¸Ê

1. ‹ŠÂ¥„˜ª´ ž¦³„°…°Š¡®œ» µ¤˜n°Åžœ¸Ê

1.1 x2 – 4 = …………………………………………………………………….

1.2 2x2-3x+1 = …………………………………………………………………….

1.3 x3 – y3 = ……………………………………………………………………..

1.4 x3 + y3 = …………………………………………………………………….

1.5 8 x3 –27 y3 = …………………………………………………………………….

1.6 2x3+3x2-2x-3 = …………………………………………………………………….

= …………………………………………………………………….

= …………………………………………………………………….
1.7 3x3-2x2-12x+8 = …………………………………………………………………….

= …………………………………………………………………….

= …………………………………………………………………….

2. ‹ŠÂ„­o ¤„µ¦˜°n ޜʸ

2.1 x2– 4 = 0 ‹³Å—o x = ……………….

2.2 2x2-3x+1 = 0 ‹³Å—o x = ……………….

2.3 2x2-3x-1 = 0 ‹³Å—o x = ……………….

2.4 3x3-2x2-27x-18 = 0 ‹³Å—o x = ……………….

­—Šª·›š¸ ε…………………………………………………………………………………………...

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

………………………………………………………………………………………………………

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

2.5 2x3-3x2-5x+6 = 0 ‹³Å—o x = ……………….

­—Šª›· š¸ ε …………………………………………………………..……………………………..

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

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

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

⌦ 47
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 6.2

‡µÎ ­ÉŠ´ ‹Š«„¹ ¬µš§¬‘¸šÂ¨³˜ª´ °¥µn Š ¨ªo ˜°‡Îµ™µ¤
1. š§¬‘¸ š˜ª´ ž¦³„°

Á¤É°º P(x) ‡°º ¡®œ» µ¤ anxn+an-1xn-1+an-2xn-2+…+a1x+a0×¥šÉ¸ n Ážœ} ‹ÎµœªœÁ˜È¤ª„ an,an-1,
…,a1,a0 Áž}œ‹Îµœªœ‹¦Š· Ž¹ÉŠ anz 0 ¡®œ» µ¤ P(x) ‹³¤¸ x-c Ážœ} ˜´ªž¦³„°„Ș°n Á¤Éº° P(c) = 0

˜´ª°¥nµŠ ‹ŠÂ­—Šªnµ x-2 Ážœ} ˜ª´ ž¦³„°…°Š¡®»œµ¤ x3+2x2-5x-6
ª›· š¸ ε Ä®o P(x) = x3+2x2-5x-6

‹µ„š§¬‘¸š˜´ªž¦³„° ¡®»œµ¤ P(x) ‹³¤¸ (x – c) Áž}œ˜´ªž¦³„°„Șn°Á¤Éº° P( c ) = 0 Ä®o

x–c =x–2
‹³Å—o c = 2
—´ŠœÊœ´ P ( 2 ) = (2)3+2(2)2-5(2)-6 = 0
Á¤ºÉ° P( 2 ) = 0 ­—Šªµn ( x – 2 ) Áž}œ˜ª´ ž¦³„°˜ª´ ®œ¹ÉŠ…°Š¡®œ» µ¤ x3+2x2-5x-6 ¨³Á¤Éº°

®µ¦ x3+2x2-5x-6 —oª¥ x – 2 ‹³Å—o Ÿ¨®µ¦ = x2+4x+3
—´ŠœÊ´œ x3+2x2-5x-6 = ( x-2) ( x2+4x+3 )

= (x-2) (x+1)(x+3)
¨³Áœ°Éº Š‹µ„ - 6 = (-2) (1) (3) —´Šœ´Êœ -2 , 1 ¨³ 3 Ážœ} ˜ª´ ž¦³„°—» ®œ¹ÉŠ…°Š -6

…°o ­Š´ Á„˜ Ĝ„¦–š¸ ´ÉªÅž ™oµ x – c Ážœ} ˜´ªž¦³„°…°Š¡®»œµ¤
anxn+ an-1xn-1+ an-2xn-2+…+ a1x+ a0 ×¥­¤´ ž¦³­š· ›…·Í °Š¡®»œµ¤
œ¸ÁÊ žœ} ‹ÎµœªœÁ˜È¤ ¨³ an= 1 ¨ªo c ‹³Ážœ} ˜ª´ ž¦³„°…°Š a0

‹µ„„µ¦«¹„¬µš§¬‘¸ šÂ¨³˜ª´ °¥µn Š…µo Š˜œo Á¤ºÉ°„ε®œ—¡®œ» µ¤ P(x) = 3x3-4x2-3x+4
1. ‹ŠÂ­—Šªnµ (x-1) Áž}œ˜ª´ ž¦³„°…°Š¡®œ» µ¤ P(x)
2. ‹ŠÂ¥„˜´ªž¦³„°…°Š P(x)

48 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 49
⌦

50 ⌫ ⌫  ⌦
 ⌫     ⌫  

1.2 2x4+3x3-16x2-8x+24 = 0
………………………………………………………………………………………………………........
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2. ™oµ¥°¤¦´ªµn 3 Áž}œ‹µÎ œªœ‹¦Š· ‹Š¡­· ‹¼ œªr µn 2+ 3 Áž}œ‹Îµœªœ°˜¦¦„¥³
………………………………………………………………………………………………………........
………………………………………………………………………………………………………........
………………………………………………………………………………………………………........
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⌦ 51
⌦

Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦¼oš¸É 7

Á¦°ºÉ Š ­¤´˜„· µ¦Å¤Án šµn „œ´ Êœ´ ¤´›¥¤«„¹ ¬µžše ɸ 4
ª· µ ‡–˜· «µ­˜¦r Áª¨µ 2 ªÉ´ äŠ

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Ÿ¨„µ¦Á¦¥¸ œ¦¼oš¸‡É µ—®ª´Š
°„­¤´˜„· µ¦Å¤nÁšnµ„œ´ ¨³œÎµÅžÄoŗo

1. ‹—» ž¦³­Š‡„r µ¦Á¦¥¸ œ¦¼o œ„´ Á¦¥¸ œ­µ¤µ¦™
1.1 °„‡ªµ¤®¤µ¥Â¨³Ä­o ´ ¨„´ ¬–„r µ¦Å¤nÁšnµ„´œÂ˜µn ŠÇ ŗo
1.2 °„­¤´˜·Å˜¦ª·£µ‡Å—o
1.3 °„­¤˜´ …· °Š„µ¦Å¤nÁšµn „´œÅ—o
1.4 œÎµ­¤˜´ …· °Š„µ¦Å¤nÁšnµ„œ´ ŞčÅo —o

2. œª‡ªµ¤‡·—®¨´„
­¤´˜·…°Š¦³‹Îµœªœ‹¦·Š 11 ž¦³„µ¦Â¦„Áž}œ¡ºÊœ“µœÄœ„µ¦Ä®o‡ªµ¤®¤µ¥…°Š„µ¦¨Â¨³

„µ¦®µ¦‹Îµœªœ‹¦·Š ¨³ÂœªšµŠµŠž¦³„µ¦Äœ„µ¦Â„o­¤„µ¦¡®»œµ¤ ­nªœ­¤´˜·ž¦³„µ¦š¸É 12 , 13
¨³ 14 Áž}œ¡Êºœ“µœÄœ„µ¦Ä®o‡ªµ¤®¤µ¥ “ œo°¥„ªnµ” ¨³ “¤µ„„ªnµ ” ¨³ÄoÁž}œÂœªšµŠÄœ„µ¦Â„o
°­¤„µ¦

3. ÁœÊ°º ®µ­µ¦³ ­¤´˜·„µ¦Å¤Án šµn „´œ

šœ·¥µ¤ 1 ­¤µ„· …°Š R Á¦¸¥„ªµn ‹µÎ œªœª„ ¨³™oµ a  R ‹³Á¦¸¥„ a ªµn
‹Îµœªœ¨

šœ·¥µ¤ 2 a < b ®¤µ¥‡ªµ¤ªnµ b a  R
a > b ®¤µ¥‡ªµ¤ªµn a b  R

52 ⌫ ⌫  ⌦
 ⌫     ⌫  

šœ¥· µ¤ 3 a d b ®¤µ¥™Š¹ a Ť¤n µ„„ªµn b ®¦º° a œo°¥„ªµn ®¦°º Ášµn „´ b
a t b ®¤µ¥™Š¹ a Ťnœ°o ¥„ªµn b ®¦º° a ¤µ„„ªnµ®¦°º Ášnµ„´ b
a b c ®¤µ¥™Š¹ a b ¨³ b c
a d b d c ®¤µ¥™¹Š a d b ¨³ b d c
a b d c ®¤µ¥™Š¹ a b ¨³ b d c
a d b c ®¤µ¥™Š¹ a d b ¨³ b c

®¤µ¥Á®˜» a d b °nµœªµn a œ°o ¥„ªnµ®¦º°Ášµn „´ b
a t b °nµœªnµ a ¤µ„„ªnµ®¦°º Ášnµ„´ b

­¤´˜Å· ˜¦ª£· µ‡ (Trichotomy property)
™oµ a ¨³ b Áž}œ‹µÎ œªœ‹¦·ŠÂ¨ªo a = b, a < b ¨³ a > b ‹³Áž}œ‹¦·ŠÁ¡¸¥Š°¥µn ŠÄ—°¥µn Š®œŠ¹É

š§¬‘¸ ššÉÁ¸ „É¥¸ ª„´­¤˜´ ·…°Š„µ¦Å¤nÁšnµ„œ´
š§¬‘¸š 1 ­¤˜´ „· µ¦™µn ¥š°— ™oµ a > b ¨³ b > c ¨oª a > c
š§¬‘¸ š 2 ­¤˜´ ·„µ¦ª„—oª¥‹µÎ œªœÁšnµ„´œ ™oµ a > b ¨ªo a+c > b+c Á¤É°º c Áž}œ‹Îµœªœ

‹¦Š· ėÇ
š§¬‘¸ š 3 ‹µÎ œªœª„¨³‹µÎ œªœ¨Áž¦¥¸ Áš¸¥„´ 0
a Áž}œ‹µÎ œªœª„ „˜È n°Á¤º°É a > 0
a Ážœ} ‹µÎ œªœ¨ „˜È n°Á¤É°º a < 0
š§¬‘¸ š 4 ­¤´˜…· °Š„µ¦‡¼–—oª¥‹µÎ œªœÁšnµ„œ´ šÉŸ ¤Án ž}œ«¼œ¥r
„¦–¸ 1 ™µo a > b ¨³ c > 0 ¨oª ac > bc
„¦–¸ 2 ™oµ a > b ¨³ c < 0 ¨ªo ac < bc
š§¬‘¸š 5 ­¤´˜·„µ¦˜—´ °°„­µÎ ®¦´„µ¦ª„
™µo a+c > b+c ¨ªo a > b
š§¬‘¸ š 6 ­¤˜´ ·„µ¦˜—´ °°„­µÎ ®¦´„µ¦‡–¼
„¦–¸ 1 ™oµ ac > bc ¨³ c > 0 ¨ªo a > b
„¦–¸ 2 ™µo ac > bc ¨³ c < 0 ¨oª a < b
š§¬‘¸š 7 ™µo r ¨³ s Ážœ} ‹Îµœªœ‹¦·Š ¨³ r < s ‹³¤‹¸ 圪œ˜¦¦„¥³ c ŽŠÉ¹ r < c < s

⌦ 53
⌦

4. „¦³ªœ„µ¦‹´—„µ¦Á¦¸¥œ¦¼o
4.1 ‡¦š¼ šªœ­¤˜´ …· °Š‹Îµœªœ‹¦Š· Á„¥¸É ª„´„µ¦ª„¨³„µ¦‡–¼ ×¥„µ¦™µ¤Ä®œo ´„Á¦¸¥œ˜°

š¸¨³‡œ Ž¹ÉŠœ´„Á¦¸¥œ‡ª¦˜°Å—oªnµ¤¸ 11 ž¦³„µ¦ ‡º° ­¤´˜·žd— („µ¦ª„, „µ¦‡¼–) „µ¦­¨´šÉ¸
(„µ¦ª„, „µ¦‡¼–) „µ¦Áž¨É¥¸ œ„¨¤n» ŗo („µ¦ª„, „µ¦‡¼–) „µ¦¤¸Á°„¨´„¬–r („µ¦ª„, „µ¦‡¼–) „µ¦¤¸
°œ· Áª°¦­r („µ¦ª„, „µ¦‡–¼ ) ¨³ „µ¦Â‹„‹Š

‡¦¼°„œ´„Á¦¸¥œªnµ ¥´Š¤¸­¤´˜·…°Š‹Îµœªœ‹¦·ŠÁ¡·É¤Á˜·¤°¸„ 3 ž¦³„µ¦ ÁœÉº°Š‹µ„ÁŽ˜…°Š‹Îµœªœ

‹¦Š· ª„ R Ážœ} ­´ÁŽ˜…°ŠÁŽ˜…°Š‹µÎ œªœ‹¦Š· ( R ) ¨³¤¸­¤˜´ · 3 ž¦³„µ¦ ŽÉ¹Š‹³Áž}œ­¤´˜·…o°šÉ¸ 12,

13 ¨³ 14 ¨³œ„´ Á¦¸¥œÁ‡¥Á¦¥¸ œ¤µÂ¨ªo (‡¦™¼ µ¤ ¨³Ä®œo ´„Á¦¸¥œnª¥„´œ˜°­¤´˜…· o° 12 ,13 ¨³ 14 )
12. 0  R ™µo a  R ¨³ a z 0 —´ŠœÊ´œ
„. a  R ®¦°º
…. a  R ( a Ážœ} ‹µÎ œªœ¨)
13. ™oµ a,b  R ¨oª a b  R
14. ™oµ a,b  R ¨oª ab  R

4.2 ššªœ­´¨´„¬–r˜nµŠÇ šÉ¸ÄoĜ„µ¦Â­—Š„µ¦Å¤nÁšnµ„´œ ץĮoœ´„Á¦¸¥œÁ˜·¤­´¨´„¬–r
>, < ®¦º° = ¦³®ªnµŠ‹Îµœªœš¸É„ε®œ—Ä®o ‡¦¼Á…¸¥œœ„¦³—µœ—ε ¨oªÄ®oœ´„Á¦¸¥œ°°„ÅžÁ˜·¤¨ŠÄœ
n°ŠªnµŠ

1. 5 …. 3 (>) 5. (-7) ….. (-9) (>)

2. –5 …. 0 (<) 6. (3+4) ….(5+2) (=)

3. 5 ….. 0 (>) 7. –10 ….(-20+30) (<)

4. -5 ….. 5 (<) 8. (-3-7) …. (-4-5) (<)

4.3 ‡¦Â¼ ‹„Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 7.1 Ä®oœ„´ Á¦¥¸ œ«¹„¬µÂ¨³˜°‡µÎ ™µ¤ Á¡Éº°­¦ž» Ážœ} œ¥· µ¤
4.4 ‡¦Â¼ ¨³œ´„Á¦¸¥œnª¥„´œÁŒ¨¥Á°„­µ¦Âœ³ÂœªšµŠš¸É 7.1 ¨³­¦»ž ŗo—´ŠœÊ¸

™oµ a ¨³ b Áž}œ‹Îµœªœ‹¦·Š­°Š‹ÎµœªœŽ¹ÉŠ a z b ¨oª a-b Áž}œ‹Îµœªœª„„Șn°Á¤ºÉ° a>b
®¦°º „¨nµª ŗoªnµ a b  R „˜È n°Á¤º°É a ! b

®¦º° a b ! 0 „Șn°Á¤ºÉ° a ! b
a-b Áž}œ‹µÎ œªœ¨„Ș°n Á¤É°º a<b ®¦°º „¨nµª ŗªo nµ a b 0 „˜È °n Á¤°ºÉ a b

4.5 ‡¦¼„ε®œ—‹Îµœªœ‹¦·ŠÄ®oœ´„Á¦¸¥œÁž¦¸¥Áš¸¥‡nµÁž}œ‡n¼Ç Ánœ 3, 5, 2, 4 Ž¹ÉŠœ´„Á¦¸¥œ‡ª¦

Áž¦¸¥Áš¸¥Å—oªµn 3 > 5 , 5 > 2 ¨³ 2 4
‡¦¼™µ¤˜n°Åžªnµ ™oµ a, b Áž}œ‹Îµœªœ‹¦·ŠÄ—Ç Á¤ºÉ°œÎµ a ¨³ b ¤µÁž¦¸¥Áš¸¥„´œÂ¨oª

‡ªµ¤­¤´ ¡´œ›šr Á¸É žœ} Şŗo ‹³Áž}œ°¥µn ŠÅ¦oµŠ Ž¹ŠÉ œ„´ Á¦¸¥œ‡ª¦˜°Å—ªo nµ
a = b ®¦°º a > b ®¦°º a < b

šÊŠ´ ­µ¤ž¦³„µ¦‹³Á„·—…œ¹Ê ¡¦°o ¤„œ´ ŗo®¦º°Å¤n
ŽÉ¹Šœ´„Á¦¸¥œ‡ª¦˜°Å—oªnµ Á„·—…ʹœÅ—oÁ¡¸¥Š°¥nµŠÁ—¸¥ª ‹³Á„·—…ʹœ¡¦o°¤„´œÅ¤nŗo ¨oª‡¦¼‹¹Š°„Ä®o
œ„´ Á¦¥¸ œš¦µªnµš„ɸ ¨nµª¤µ…µo Š˜oœÁ¦µ‹³Á¦¸¥„ªµn “­¤˜´ „· µ¦Áž}œ°¥µn Š®œÉŠ¹ Ĝ­µ¤” ®¦º° ­¤´˜·Å˜¦ª·£µ‡

(Trichotomy Property)

54 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.6 Ä®oœ„´ Á¦¥¸ œšÎµÂ „f ®´— 2.7 …o° 1-9 Ĝ®œŠ´ ­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡·É¤Á˜·¤ ‡–·˜«µ­˜¦r
Á¨¤n 1 …°Š ­­ªš.

4.7 ‡¦¼°›·µ¥Â f„®´— 2.7 …o°š¸Éœ´„Á¦¸¥œ­Š­´¥®¦º°­nªœÄ®nšÎµŸ·— Ž¹ÉŠ—¼‹µ„„µ¦˜¦ª‹
 „f ®—´

4.8 ‡¦¼ÂnŠ„¨n»¤œ´„Á¦¸¥œÃ—¥Ä®o‹´‡n¼„´œ ¨oªÂ‹„Á°„­µ¦Âœ³ÂœªšµŠš¸É 7.2 Ä®onª¥„´œ
˜°‡Îµ™µ¤

4.9 Ä®oœ´„Á¦¸¥œnª¥„´œÁŒ¨¥‡Îµ˜°Á°„­µ¦Âœ³ÂœªšµŠ 7.2 „¨n»¤¨³ 1 …o° ×¥‡¦¼Â¨³Á¡Éº°œÇ
nª¥„´œ ˜¦ª‹­°‡ªµ¤™¼„˜o°Š

4.10 ‡¦¼ °„œ´„Á¦¥¸ œªµn š´ÊŠ®¤—š„ɸ ¨µn ª¤µÂ¨ªo Á¦¥¸ „ªnµ ­¤´˜·„µ¦Å¤Án šµn „œ´
4.11 ‡¦¼Â¨³œ´„Á¦¸¥œnª¥„´œ­¦»ž ­¤´˜·„µ¦Å¤nÁšnµ„´œ ¨³œ´„Á¦¸¥œ‹—´œš¹„ ¡¦o°¤š´ÊŠ
¥„˜´ª°¥µn Šž¦³„° ŗ—o ´ŠœÊ¸

­¤´˜…· °Š„µ¦Å¤nÁšnµ„œ´

­¤˜´ „· µ¦Å¤nÁšnµ„´œ ˜´ª°¥µn Š
1. ­¤´˜·„µ¦™µn ¥š°—
4 > 3 ¨³ 3 > 2 ¨oª 4 > 2
™oµ a > b ¨³ b > c ¨ªo a > c
2. ­¤´˜„· µ¦ª„—oª¥‹ÎµœªœÁ—¸¥ª„´œ 3 > 2 —´Šœœ´Ê 3+1 > 2+1 ‡º° 4 > 3

™µo a > b ¨ªo a+c > b+c 4 > 3 ¨³ 2>0 —´Šœœ´Ê 4(2)>3(2) ®¦°º 8>6
3. ­¤´˜„· µ¦‡–¼ —oª¥‹ÎµœªœÁ—¥¸ ª„œ´ 4 > 3 ¨³ –2<0 —´ŠœÊœ´ 4(-2)<3(-2) ®¦°º -8<-6

3.1 ™µo a > b ¨³ c > 0 ¨oª ac > bc 4+2 > 3+2 ¨oª 4 > 3
3.2 ™µo a > b ¨³ c < 0 ¨ªo ac < bc
4. ­¤˜´ ·„µ¦˜—´ °°„­µÎ ®¦´ „µ¦ª„ 4(2) > 3(2) ¨ªo 4 > 3 ×¥ 2 > 0
™µo a+c > b+c ¨oª a > b 4(-2) > 5(-2) ¨oª 4 < 5 ×¥ –2 < 0
5. ­¤˜´ ·„µ¦˜´—°°„­Îµ®¦´ „µ¦‡–¼
5.1 ™oµ ac > bc ¨³ c > 0 ¨oª a > b
5.2 ™oµ ac > bc ¨³ c < 0 ¨oª a < b

4.12 ‹µ„­¤˜´ ·„µ¦Å¤Án šµn „œ´ ‡¦¼ °„œ„´ Á¦¥¸ œªµn Á¦µ ¥Š´ ¤¸œ·¥µ¤Á¡É·¤Á˜¤· —Š´ œ¸Ê

šœ·¥µ¤
a d b ®¤µ¥™¹Š a Ť¤n µ„„ªµn b
a t b ®¤µ¥™Š¹ a Ťnœo°¥„ªnµ b
a b c ®¤µ¥™Š¹ a b ¨³ b c
a d b d c ®¤µ¥™Š¹ a d b ¨³ b d c
a b d c ®¤µ¥™Š¹ a b ¨³ b d c

®¤µ¥Á®˜» a d b °µn œªµn a œ°o ¥„ªnµ®¦°º Ášµn „´ b
a t b °nµœªnµ a ¤µ„„ªµn ®¦°º Ášµn „´ b

⌦ 55
⌦

4.13 Ä®oœ„´ Á¦¥¸ œšµÎ Á°„­µ¦ f„®—´ šÉ¸ 7
4.14 ‡¦¼ÁŒ¨¥Á°„­µ¦ „f ®´—šÉ¸ 7
4.15 Ä®œo „´ Á¦¥¸ œ«„¹ ¬µÁ°„­µ¦‡ªµ¤¦oš¼ ɸ 7 Á¡¤É· Á˜¤· ×¥‡¦¼°›· µ¥Á¡¸¥ŠÁ¨È„œo°¥ ¨ªo Ä®œo „´ Á¦¥¸ œ
Ş«¹„¬µ¤µÁžœ} „µ¦oµœ

5. ®¨nŠ„µ¦Á¦¸¥œ¦o¼
5.1 Á°„­µ¦Âœ³ÂœªšµŠš¸É 7.1 , 7.2
5.2 Á°„­µ¦ f„®—´ šÉ¸ 7 Á°„­µ¦‡ªµ¤¦šo¼ ¸É 7 (Á¡·É¤Á˜·¤)
5.3 ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦¼Áo ¡É¤· Á˜·¤ ‡–˜· «µ­˜¦r Á¨n¤ 1 Êœ´ ¤´›¥¤«„¹ ¬µžešÉ¸ 4

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤œ· Ÿ¨

1. ­Š´ Á„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™„¼ ˜°o ŠÁžœ} ­nªœ¤µ„

2. ­Š´ Á„˜‹µ„„µ¦¦ªn ¤„·‹„¦¦¤ 2. œ„´ Á¦¥¸ œ¦nª¤„·‹„¦¦¤—¸

3. ­´ŠÁ„˜‹µ„‡ªµ¤­œÄ‹ 3. œ´„Á¦¸¥œ˜Ê´ŠÄ‹Á¦¸¥œ—¸ ¨³Ä®o‡ªµ¤­œÄ‹

4. šµÎ ˚¥rÁ°„­µ¦Âœ³ÂœªšµŠ 7.1,7.2 ­¤ÉµÎ Á­¤°

¨³Á°„­µ¦ f„®´—š¸É 7 4. œ´„Á¦¥¸ œšÎµÅ—™o „¼ ˜o°Šž¦³¤µ– 85 %

5. šÎµÃ‹š¥rĜ®œŠ´ ­°º Á¦¸¥œÂ „f ®´— 5. œ´„Á¦¥¸ œšÎµÅ—™o „¼ ˜°o Šž¦³¤µ– 80 %

2.4 …o° 1-7

7. ´œš„¹ ®¨Š´ ­°œ
....................................................................................................................................................................
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....................................................................................................................................................................
....................................................................................................................................................................

8. „‹· „¦¦¤Á­œ°Âœ³
....................................................................................................................................................................
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56 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 7.1

‡Îµ­´ÉŠ ‹Š®µŸ¨¨ ×¥Á˜·¤‡Îµ˜°¨ŠÄœ°n Šªµn Š

…o° ˜ª´ ˜Š´Ê ˜´ª¨ Ÿ¨¨ ‹µÎ œªœª„®¦°º ‹Îµœªœ¨

15 2

26 -3

3 -5 -7

44 9

5 -2 8

6 -8 -5

‹µ„…°o 1. 5-2 = 3 œœ´É ‡°º 5-2 Áž}œ‹ÎµœªœÁ˜È¤ª„

2. 6-(-3) = ………. œœÉ´ ‡º° 6-(-3) Ážœ} ‹µÎ œªœÁ˜¤È ª„

3. –5-(-7) = ………. œ´Éœ‡º° –5-(-7) Áž}œ‹µÎ œªœÁ˜¤È ª„

…°o ­Š´ Á„˜ ‹µ„…o° 1-3 ˜´ª˜ÊŠ´ ¤µ„„ªµn ˜ª´ ¨ Ÿ¨¨Ážœ} ‹Îµœªœª„

—´Šœœ´Ê ™oµ a > b ¨oª a - b Ážœ} ‹Îµœªœ ……….
‹µ„…o° 4. 4-9 = ………. œÉœ´ ‡°º 4-9 Ážœ} ‹µÎ œªœ ……….

5. –2-8 = ………. œ´Éœ‡°º –2-8 Áž}œ‹µÎ œªœ ……….
6. –8-(-5) = ………. œœÉ´ ‡º° –8-(-5) Ážœ} ‹µÎ œªœ ……….

…o°­Š´ Á„˜ ‹µ„…o° 4 - 6 ˜ª´ ˜ÊŠ´ ………. ˜´ª¨ Ÿ¨¨Áž}œ‹µÎ œªœ ……….
—Š´ œœÊ´ ™µo a < b ¨oª a-b Ážœ} ‹µÎ œªœ ……….

…wwwwwww…

⌦ 57
⌦

Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 7.2

‡Îµ­ÉŠ´ ‹ŠÁ˜¤· ‡µÎ ˜°¨ŠÄœ°n ŠªnµŠÄ®o™„¼ ˜o°Š­¤¦¼ –r
1. 5 >3 ¨³ 3 > 2 —Š´ œ´Êœ­¦»žÅ—oªnµ 5 >2
2. –2 >-5 ¨³ –5 >-8 —´ŠœœÊ´ ……….………
3. ™µo a >b ¨³ b >c ¨ªo ……….……….
4. ™µo 6 >4 Á¤Éº°œµÎ 2 ª„šÊ´Š 2 …µo Š ‹³Å—o (6+2) ………. (4+2)
5. 10 >7 œÎµ (-5) ª„šÊŠ´ 2 …oµŠ ‹³Å—o ……….……….
6. ™µo a > b ¨oª œµÎ c ª„šÊŠ´ 2 …µo Š ‹³Å—o ……….……….
7. 5 >3 œµÎ 2 ‡¼–šŠÊ´ 2 …µo Š ‹³Å—o (5¯2) ………. (3¯2)
8. –4 >-6 œÎµ (-2) ‡–¼ š´ŠÊ 2 …oµŠ ‹³Å—o ……….……….
9. ™oµ a >b ¨³ c >0 —Š´ œ´Êœ ac ………. bc

™oµ a >b ¨³ c <0 —Š´ œÊœ´ ac ………. bc
10. ™oµ 5+2 >3+2 œµÎ (-2) ª„šŠ´Ê 2 …µo Š ‹³Å—o ……….……….
11. ™µo 6+(-3) >4+(-3) œµÎ 3 ª„šŠÊ´ 2 …oµŠ ‹³Å—o ……….……….
12. ™µo a+c >b+c ¨ªo ……….……….
13. 8(2) >5(2) œÎµ 1 ‡¼–š´ÊŠ 2 …oµŠ ‹³Å—o ……….……….

2

14. 8(-3) >10(-3) œÎµ 1 ‡¼–šÊ´Š 2 …oµŠ ‹³Å—o ……….……….

3

15. ™µo ac >bc ¨³ c >0 ¨oª‹³Å—o ……….……….
™oµ ac >bc ¨³ c <0 ¨oª‹³Å—o ……….……….

58 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®—´ š¸É 7

1. …o°‡ªµ¤˜n°ÅžœÊ¸Ážœ} ‹¦Š· Á­¤° ‹Š°„ªnµÄ­o ¤˜´ °· ³Å¦Äœ„µ¦Ä®oÁ®˜»Ÿ¨

1.1 5<7 ¨³ 7<9 —´ŠœœÊ´ 5<9 ……….……….

1.2 a>b —´Šœ´Êœ a+1>b+1 ……….……….

1.3 –4<3 —Š´ œÊ´œ 2(-4)<2(3) ……….……….

1.4 –5<-2 —Š´ œ´œÊ (–5)(-3)>(-2)(-3) ……….……….

1.5 –10<15 —Š´ œ´œÊ 10 §¨ 1 ¸· ! 12 §¨ 1 ¸· ……….……….
2 ¹ 2 ¹
© ©

2. …°o ‡ªµ¤š„¸É µÎ ®œ—Ä®˜o °n Şœ¸Ê ‹¦·Š®¦º°ÁšÈ‹
2.1 6 > 4 —Š´ œ´Êœ 6+(-1) > 4+(-1)
2.2 5 < 6 —´ŠœÊ´œ 5(-2) < 6(-2)
2.3 –4 < 8 —´Šœ´Êœ 2 >-4
2.4 3 >0 ¨³ –5 < 0 —Š´ œœ´Ê (3)(-5) < 0
2.5 ™µo a > b ¨ªo a2 > b2
2.6 ™µo a2 > b2 ¨oª a > b
2.7 ™oµ 0 < a < b ¨oª 1 ! 1

ab

2.8 ™oµ a < b < 0 ¨ªo 1 ! 1

ab

2.9 ™µo a < b ¨³ c < d ¨oª a+c < b+d
2.10 ™µo a < b ¨³ c < d ¨ªo ac < bd

⌦ 59
⌦

Á°„­µ¦‡ªµ¤¦šo¼ ɸ 7

‡µÎ Â¸Ê ‹Š ‹Š«¹„¬µ…o°‡ªµ¤˜°n ޜʸ¨ªo šµÎ  f„®´—
œ°„‹µ„­¤´˜„· µ¦Å¤nÁšµn „œ´ ˜µ¤š¸„É ¨µn ª¤µÂ¨ªo ¦³‹Îµœªœ‹¦Š· ¥´Š¤¸­¤´˜·„µ¦Å¤Án šnµ„´œ

—Š´ ˜°n Şœ¸Ê
1. ™µo a<b ¨³ c<d ¨oª

1.1 a+c<b+d

1.2 a-d<b-c
2. ™µo 0<a<b ¨³ 0<c<d ¨oª ac<bd
3. ™µo a<b<0 ¨³ c<d<0 ¨oª ac>bd
4. ™µo 0<a<b ¨³ 0<c<d ¨ªo a b

dc

5. ™µo a<b<0 ¨³ c<d<0 ¨ªo a ! b

dc

˜´ª°¥µn Š ™oµ 2 < x < 8 ¨³ 1 < y < 5 ‹Š®µ x+y, x-y, xy ¨³ x
1) ®µ x + y
y

2) ®µ x – y

2 < x < 8 ¤‡¸ ªµ¤®¤µ¥Á®¤°º œ 2 < x ¨³ x < 8 2 < x ¨³ x < 8

1 < y < 5 ¤‡¸ ªµ¤®¤µ¥Á®¤°º œ 1 < y ¨³ y < 5 -1 > - y ¨³ -y > -5

? 3 < x + y ¨³ x + y < 13 -5 < - y ¨³ -y < -1

—Š´ œ´Êœ 3 < x + y < 13 ? -3 < x - y ¨³ x - y < 7

—Š´ œÊœ´ -3 < x - y < 7

* ­nªœ xy ¨³ x Ä®oœ„´ Á¦¥¸ œnª¥„œ´ ®µ‡µÎ ˜°

y

Á°„­µ¦ „f ®—´
1. ™µo 3 < x < 10 ¨³ 1 < y < 6 ‹Š®µ

1. x-y 2. x2+y 3. x2y2 x

2. ™µo 4 < x < 6 ¨³ 2 < y < 5 ‹Š®µ 4.

1. x2+y2 2. x2-y 3. xy y
y

4.

x

60 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 61
⌦

62 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 63
⌦

šÎµÂ „f ®—´ 2.5 „ …o° 2,3 Ĝ®œŠ´ ­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Á¼o ¡·É¤Á˜¤· ² Áž}œ„µ¦µo œ
5. ®¨nŠ„µ¦Á¦¸¥œ¦o¼

5.1 Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 8
5.2 Á°„­µ¦ „f ®—´ šÉ¸ 8.1 , 8.2
5.3 ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦¼oÁ¡É¤· Á˜¤· ‡–·˜«µ­˜¦r Á¨¤n 1 œ´Ê ¤›´ ¥¤«„¹ ¬µžše ¸É 4

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª´—Ÿ¨ „µ¦ž¦³Á¤·œŸ¨

1. ­Š´ Á„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™„¼ ˜°o ŠÁž}œ­ªn œ¤µ„

2. ­´ŠÁ„˜‹µ„„µ¦¦nª¤„·‹„¦¦¤ 2. œ´„Á¦¸¥œ¦nª¤„·‹„¦¦¤—¸

3. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 3. œ´„Á¦¥¸ œ˜ŠÊ´ ċÁ¦¥¸ œ—¸Â¨³Ä®o‡ªµ¤­œÄ‹

4. šÎµÃ‹š¥rÁ°„­µ¦Âœ³ÂœªšµŠ 8 ­¤ÎɵÁ­¤°

¨³Á°„­µ¦ f„®´— 8.1 , 8.2 4. œ„´ Á¦¥¸ œšµÎ ŗo™¼„˜o°Šž¦³¤µ– 85 %

5. šµÎ ˚¥r  „f ®—´ 2.5 „ …°o 2,3

Ĝ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦oÁ¼ ¡¤·É Á˜¤· ² 5. œ´„Á¦¸¥œšµÎ ŗo™„¼ ˜o°Šž¦³¤µ– 90 %

7. œ´ š„¹ ®¨Š´ ­°œ
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8. „‹· „¦¦¤Á­œ°Âœ³
....................................................................................................................................................................
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64 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 65
⌦

Á°„­µ¦ f„®—´ 8.1

1. ‹ŠÁ…¥¸ œÂ­—Šªn Šš¸„É µÎ ®œ—˜n°ÅžœÊ¸œÁ­œo ‹µÎ œªœ

1. (-1,3) 2. [-1,3] 3. (-2,1] 4. [-3,2]
5. (-2,f) 6. [-2, f) 7. (-f,2) 8. (-f,2]

1. 5.
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

2. 6.
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

3. 7. -3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3

4. 8.
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

2. ‹ŠÁ…¸¥œªn Š˜°n ޜʸĮo°¥nļ œ¦ž¼ …°ŠÁŽ˜

nªŠ ‡Îµ˜°Äœ¦ž¼ ÁŽ˜

(-3,3) ………..………..………..………..………..………..………..……

(-1,5] ………..………..………..………..………..………..………..……

[-5,0] ………..………..………..………..………..………..………..……
(-f,5] ………..………..………..………..………..………..………..……
[2, f) ………..………..………..………..………..………..………..……
(-f,0) ………..………..………..………..………..………..………..……
[-5,2) ………..………..………..………..………..………..………..……
(-f,2] ………..………..………..………..………..………..………..……

3. ‹ŠÁ…¥¸ œÁŽ˜˜n°ÅžœÊĸ ®o°¥nļ œ¦ž¼ …°ŠnªŠ

ÁŽ˜ ‡µÎ ˜°Äœ¦¼žªn Š
{x| 3 xd5 } ………..………..………..………..………..………..………..…….
{x| -3dxd2 } ………..………..………..………..………..………..………..…….
{x| -2 x 3 } ………..………..………..………..………..………..………..…….
{x| -3dx 5 } ………..………..………..………..………..………..………..…….
{x| x!2 } ………..………..………..………..………..………..………..…….
{x| xt-6 } ………..………..………..………..………..………..………..…….
{x| x -3 } ………..………..………..………..………..………..………..…….
{x| xd4 } ………..………..………..………..………..………..………..…….

66 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ f„®´—š¸É 8.2

1. Ä®o A=(3,6) ¨³ B=[4,7] ‹Š®µ
B

A

123456789

1. A ‰ B = ……….……….……….……. 2. A ˆ B = ……….……….……………..
3. Ac = …….……….……….………. 4. Bc = ……….……….……………..
5. A – B = ……….…….……….………. 6. B – A = ……….……….………...........

2. „ε®œ—Ä®o A = {xR | -4 x 4 } C
B = {xR | -8dx 2 }
C = {xR | 0dxd10 }

B
A

- - -8 -6 -4 -2 0 2 4 6 8 10 12

3. ‹Š®µÁŽ˜˜°n Şœ¸Ê (˜°Äœ¦ž¼ nªŠ) 2. A ˆ B = ……….……….…………...
1. A ‰ B = ……….……….…………… 4. A ˆ Bc = ……….……….……………
3. (A ‰ B) ˆ C = ……….……….…… 6. Cc ‰ B = ……….……….……………
5. Ac ˆ B = ……….……….…………..

4. ‹Š®µÁŽ˜‡Îµ˜°‹µ„nªŠš¸„É µÎ ®œ—Ä®o˜n°Åžœ¸Ê 6. (3,5] – (2,5] = …...…. …
1. [2,6] ‰ [1,4] = [1,6] = ………….
2. (1,3) ‰ {1,3} = ………… 7. [2,5] – (1,3) = ……...…..
3. [3,7] ˆ (1,4) = ………... = ………....
4. (3,f) ˆ [7,12) = ….…….. 8. [2,5) = ……..…..
5. (1,4) ˆ (4,6) = …….….. 9. (-f,4)c
10. [6,f)c

⌦ 67
⌦

Ÿœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦o¼šÉ¸ 9

Á¦º°É Š „µ¦Â„°o ­¤„µ¦ Ê´œ¤›´ ¥¤«„¹ ¬µžše ɸ 4
ª· µ ‡–˜· «µ­˜¦r Áª¨µ 2 ´ÉªÃ¤Š

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Ÿ¨„µ¦Á¦¸¥œ¦š¼o ¸‡É µ—®ªŠ´
„o°­¤„µ¦˜´ªÂž¦Á—¥¸ ªÅ—o

1. ‹—» ž¦³­Š‡r„µ¦Á¦¥¸ œ¦¼o œ´„Á¦¥¸ œ­µ¤µ¦™
1.1 œµÎ ‡ªµ¤¦Áo¼ ¦Éº°ŠnªŠ¤µÄoŗo
1.2 œÎµ­¤´˜·„µ¦Å¤nÁšµn „œ´ …°Š‹µÎ œªœ‹¦Š· ¤µÄoĜ„µ¦Â„°o ­¤„µ¦˜´ªÂž¦Á—¸¥ªÅ—o
1.3 „°o ­¤„µ¦˜´ªÂž¦Á—¥¸ ªÅ—o

2. œª‡ªµ¤‡·—®¨´„
°­¤„µ¦Äœ x Ážœ} ž¦³Ã¥‡š¸É¤˜¸ ´ªÂž¦ x ¨³„¨µn ª™Š¹ „µ¦Å¤nÁšnµ„œ´ Áœn 2x < 8 , 2x z 8 ,

3x+1 t5
°­¤„µ¦‹³Áž}œ‹¦Š· ®¦°º ÁšÈ‹…œÊ¹ °¥¼n„´‹Îµœªœ‹¦Š· šœÉ¸ ε¤µÂšœšÉ¸˜´ªÂž¦Äœ°­¤„µ¦ÁŽ˜‡Îµ˜°…°Š

°­¤„µ¦Äœ x Áž}œÁŽ˜šÉ¸¤¸­¤µ·„Áž}œ‹Îµœªœ‹¦·Š ×¥šÉ¸‹ÎµœªœÁ®¨nµœÊ´œÁ¤ºÉ°œÎµ¤µÂšœ x ¨oªšÎµÄ®o
°­¤„µ¦Áž}œ‹¦Š· „µ¦Â„°o ­¤„µ¦‡°º „µ¦®µÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦

3. Áœ°ºÊ ®µ­µ¦³ „µ¦Â„°o ­¤„µ¦˜ª´ ž¦Á—¸¥ª

˜ª´ °¥µn Šš¸É 1 ‹ŠÂ„°o ­¤„µ¦ 5 d 7-2x d 11

ª·›¸šµÎ ‹µ„ 5 d 7-2x d 11

‹³Å—o 5 d 7-2x ¨³ 7-2x d 11

ª„—ªo ¥ –7 šÊ´Š 2 …µo Š ; -2 d -2x ¨³ -2x d 4

‡–¼ —oª¥ 1 šŠ´Ê 2 …oµŠ ; 1tx ¨³ x t -2
2

œÉœ´ ‡°º x d 1 ¨³ x t -2

ŽŠ¹É ­—Šª·›¸®µ‡Îµ˜°—ªo ¥„¦µ¢Å——o Š´ œÊ¸

-2 -1 0 1 2
‹³Å—Áo Ž˜‡Îµ˜°…°Š°­¤„µ¦‡°º {xR | -2 d x d 1} ®¦º° [-2,1]

68 ⌫ ⌫  ⌦
 ⌫     ⌫  

˜´ª°¥µn Šš¸É 2 ‹Š®µÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦ 9-2x 4x-3 d x+6
ª›· ¸šÎµ ‹µ„ 9-2x 4x-3 d x+6
‹³Å—o 9-2x 4x-3 ¨³ 4x-3 d x+6
12 6x ¨³ 3x d 9
2 x ¨³ x d 3

23
œ´Éœ‡º° 2 x d 3
ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡º° {xR | 2 x d 3} ®¦º° (2,3]

˜´ª°¥µn Šš¸É 3 ‹Š®µÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ x2 4x 3 ! 0
ª·›š¸ µÎ ‹µ„°­¤„µ¦ x2 4x 3 ! 0
‹³Å—o (x-1)(x-3) > 0
„¦–¸šÉ¸ 1 ™oµ (x-1) > 0 ¨³ (x-3) > 0
‹³Å—o x > 1 ¨³ x > 3
œ´Éœ‡º° x > 3
„¦–š¸ ɸ 2 ™oµ (x-1) <0 ¨³ (x-3)< 0
‹³Å—o x < 1 ¨³ x < 3
œœÉ´ ‡º° x < 1 œœÉ´ Á°Š

ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡º° {xR | x ! 3 ®¦º° x 1} = (3,f) ‰ ( f,1)

˜ª´ °¥µn Šš¸É 4 ‹ŠÂ„°o ­¤„µ¦ x2 2x d 3
ª›· š¸ µÎ ‹µ„ °­¤„µ¦ x2 2x d 3

(x 3)(x 1) d 0

„¦–¸šÉ¸ 1 ™oµ (x-3) t 0 ¨³ (x+1) d0

‹³Å—o x t 3 ¨³ xd-1

ŽÉŠ¹ Ť¤n ¸‡µn x ėšÉ¸¤µ„„ªµn ®¦°º Ášnµ„´ 3 ¨³œ°o ¥„ªµn ®¦°º Ášµn „´ 1

„¦–¸šÉ¸ 2 ™µo (x-3) d0 ¨³ (x+1) t0

‹³Å—o xd3 ¨³ xt -1

ϫσ ࡼ -1 d x d 3

—Š´ œÊœ´ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡°º { xR | -1 d x d 3 }

⌦ 69
⌦

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¸¥œ¦o¼
4.1 ‡¦Á¼ Œ¨¥Á°„­µ¦ „f ®—´ šÉ¸ 8.2 ¨³Â f„®—´ 2.5 ÁŒ¡µ³…°o šœÉ¸ „´ Á¦¸¥œ­Š­¥´ ®¦°º …°o šœ¸É ´„Á¦¥¸ œ

­nªœÄ®nšÎµŸ—· Ž¹ÉŠ¡‹µ„„µ¦˜¦ª‹Â „f ®—´

‡¦¼ššªœ­¤˜´ ·…°Š„µ¦Å¤nÁšµn „œ´ ×¥„µ¦˜Š´Ê ‡µÎ ™µ¤Ä®oœ´„Á¦¥¸ œ˜° ×¥Áœœo ­¤´˜·„µ¦‡¼–

—ªo ¥‹µÎ œªœÁ—¥¸ ª„´œ ¨³„µ¦˜´—°°„­µÎ ®¦´„µ¦‡–¼ Á¤ºÉ°‹µÎ œªœšœ¸É µÎ ¤µ‡¼–®¦°º ˜—´ °°„œo°¥„ªnµ 0 ŽÉŠ¹ ‹³

šÎµÄ®Áo ‡¦°ºÉ Š®¤µ¥…°Š°­¤„µ¦Áž¨¸É¥œÁž}œ˜¦Š„´œ…oµ¤ „¨nµª‡º° ‹µ„ ! Áž}œ ®¦º°‹µ„ Ážœ} ! Áœn

„. ™µo 5 > 2 ¨oª 2 ˜5 2 ˜2

…. ™oµ 10x ! 30 ¨oª §¨© 1 ·¹¸ 10 x ©¨§ 1 ¹·¸ 30
10 10

®¦º° x 3

4.2 ‡¦Â¼ ‹„Á°„­µ¦ f„®—´ šÉ¸ 9 Ä®oœ„´ Á¦¥¸ œš»„‡œšµÎ Á¡°ºÉ Ážœ} „µ¦ššªœ„µ¦Â„o°­¤„µ¦—„¸ ¦¸ 1

Ž¹ÉŠœ„´ Á¦¥¸ œÁ‡¥Á¦¥¸ œ¤µÂ¨ªo Ĝ œÊ´ ¤.3 čoÁª¨µž¦³¤µ– 10 œµš¸ ¨oª‡¦¼ÁŒ¨¥‡µÎ ˜°œ„¦³—µœ—ε
4.3 ‡¦¼¥„˜´ª°¥nµŠÂ¨³Â­—Šª›· ¸šµÎ ×¥˜Ê´Š‡µÎ ™µ¤Ä®oœ„´ Á¦¸¥œ˜°š»„…œÊ´ ˜°œ—´Šœ¸Ê

˜´ª°¥µn ŠšÉ¸ 1 ‹Š®µÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ 5 d 7-2x d 11

ª›· ¸šÎµ ×¥°µ«¥´ ‡ªµ¤¦o¼šª¸É µn ™µo a x b ‹³Å—o a x ¨³ x b

‹µ„ 5 d 7-2x d 11

‹³Å—o 5 d 7-2x ¨³ 7-2x d 11

ª„—ªo ¥ –7 šÊŠ´ 2 …oµŠ ; -2 d -2x ¨³ -2x d 4

‡¼–—ªo ¥ 1 šÊ´Š 2 …µo Š 1 t x ¨³ x t -2
2

œœ´É ‡º° x d 1 ¨³ x t -2

ŽŠÉ¹ ­—Šª›· ®¸ µ‡µÎ ˜°Ã—¥„¦µ¢Å——o Š´ œÊ¸

-2 -1 0 1 2
‹³Å—Áo Ž˜‡µÎ ˜°…°Š°­¤„µ¦‡º° {x| -2 d x d 1} ®¦º° [-2,1]

˜ª´ °¥µn ŠšÉ¸ 2 ‹Š®µÁŽ˜‡Îµ˜°…°Š­¤„µ¦ 9-2x 4x-3 d x+6
‹µ„ 9-2x 4x-3 d x+6
‹³Å—o 9-2x 4x-3 ¨³ 4x-3 d x+6
12 6x ¨³ 3x d 9
2 x ¨³ x d 3

23
ϫσ ༡ 2 x d 3
ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡°º {x| 2 x d 3} ®¦º° (2,3]

70 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.4 ‡¦¼Á…¥¸ œÃ‹š¥r œ„¦³—µœ 1 …o° Ä®œo „´ Á¦¸¥œšµÎ 榚µÎ ŗoÄ®°o °„¤µÂ­—Šª·›¸šÎµœ„¦³—µœ
‡¦Â¼ ¨³Á¡É°º œÇ nª¥„´œ˜¦ª‹ (˜´ª°¥nµŠÃ‹š¥r Áœn ‹Š®µÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦ 3 x - 5 < x + 1 < x +

7 < 5 x + 11

4.5 ‡¦°¼ ›·µ¥­É·Šš¸Éœ„´ Á¦¸¥œ­Š­¥´ ‹œ„¦³šŠ´É Á…oµÄ‹—¸ ¨ªo Ä®šo 坝 „f ®´— 2.5 … …o° 1- 4 Ĝ®œ´Š­º°

Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦Áo¼ ¡É·¤Á˜¤· ² Áž}œ„µ¦oµœ

4.6 ‡¦š¼ šªœÁ¦ºÉ°Š„µ¦Â¥„˜ª´ ž¦³„°…°Š¡®œ» µ¤—„¸ ¦­¸ °Š Ĝ¦¼ž ax2+ bx + c ¨³Äœ¦¼žŸ¨˜µn Š

„ε¨´Š­°Š „µÎ ¨Š´ ­°Š­¤¼¦–r ץĮœo „´ Á¦¸¥œnª¥„œ´ Â¥„˜´ªž¦³„°…°Š¡®œ» µ¤˜°n ޜʸ (ªŠÁ¨È …oµŠ

®¨Š´ ‡ª¦Áž}œ‡µÎ ˜°šœÉ¸ ´„Á¦¸¥œÂ¥„Å—)o = …..…..…………. x 5 ˜ x 2
1. x2 3x 10 = …..…..………….
2. x 2 5x 4 = …..…..…………. x 1 ˜ x 4
3. 2x2 3x 1 = …..…..…………. 2x 1 ˜ x 1
4. 3x 2 5x 2 = …..…..…………. 3x 1 ˜ x 2
5. x2 16 = …..…..…………. x 4 ˜ x 4
6. 4x2 25 2x 5 ˜ 2x 5

7. x2 4x 4 = …..…..…………. x 2 2
8. x 2 6x 7
= …..…..…………. x 3 2 ˜ x 3 2

4.7 ‡¦Ä¼ ®˜o ª´ °¥µn Šš¸É 3 ¨³ 4 Ážœ} °­¤„µ¦Äœ¦¼ž„ε¨Š´ ­°Š ­—Šª›· š¸ εץč„o µ¦™µ¤˜°
Ä®œo „´ Á¦¸¥œÅ—Âo ­—Š‡ªµ¤‡·—Á®œÈ š„» …œ´Ê ˜°œÄœ„µ¦šÎµ ™µo ¤œ¸ ´„Á¦¸¥œ‡œÄ—ŤnÁ…µo ċ ‹³°›· µ¥ŽÊε°„¸ ‡¦ŠÊ´

˜ª´ °¥nµŠš¸É 3 ‹ŠÂ„o°­¤„µ¦ x2 4x 3 ! 0
ª›· ¸šÎµ ‹µ„°­¤„µ¦ x2 4x 3 ! 0
‹³Å—o (x-1)(x-3) > 0

Ÿ¨‡¼–…°Š‹ÎµœªœÄœªŠÁ¨Èš´ÊŠ­°Š¤µ„„ªnµ 0 ŽÉ¹Š®¤µ¥™¹Š Áž}œ‹Îµœªœª„
­—Šªnµ Áž}œ‹µÎ œªœª„š´ÊŠ‡¼n ®¦°º ¨šÊ´Š‡n¼ ‹¹Š¡‹· µ¦–µÁž}œ 2 „¦–¸ ‡°º

„¦–š¸ ɸ 1 ™µo (x-1) > 0 ¨³ (x-3) > 0 (Ážœ} ª„šÊ´Š‡)n¼
‹³Å—o x > 1 ¨³ x > 3 Ĝ…–³Á—¥¸ ª„œ´
œÉ´œ‡º° x > 3 ®¦º°Á…¸¥œÁž}œnªŠ‡Îµ˜°Å—oÁž}œ (1,f) ˆ (3,f) =

(3,f) Á…¥¸ œÂ­—Š—ªo ¥„¦µ¢œÁ­œo ‹µÎ œªœÅ——o Š´ œ¸Ê

13

⌦ 71
⌦

„¦–¸š¸É 2 ™µo (x-1) < 0 ¨³ (x-3) < 0 (Áž}œ¨šÊŠ´ ‡)n¼
‹³Å—o x < 1 ¨³ x < 3

‡µn x šœÉ¸ °o ¥„ªµn 1 ¨³œ°o ¥„ªµn 3 Ĝ…–³Á—¸¥ª„œ´ ‡°º x < 1 œÉœ´ Á°Š
®¦º°Á…¥¸ œÁžœ} ªn Š‡µÎ ˜°Å—o ( f,1) ˆ ( f,3) = ( f,1) ŽŠÉ¹ Á…¥¸ œÂ­—Š—oª¥„¦µ¢œÁ­œo ‹Îµœªœ
ŗ—o Š´ œÊ¸

13

—´ŠœÊ´œ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡º°¥¼Áœ¸¥œ…°ŠÁŽ˜‡Îµ˜°…°Š „¦–¸šÉ¸ 1 „´ÁŽ˜‡Îµ˜°…°Š „¦–¸š¸É 2
œœÉ´ Á°Š ‡º° (3,f) ‰ ( f,1)

®¦°º ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡°º {x  R | x <1 ®¦°º x ! 3}
Á…¸¥œÂ­—Š—oª¥„¦µ¢œÁ­œo ‹ÎµœªœÅ——o Š´ œ¸Ê

13

˜´ª°¥µn ŠšÉ¸ 4 ‹ŠÂ„°o ­¤„µ¦ x2 2x d 3
‹µ„ °­¤„µ¦ x2 2x d 3
‹³Å—o x2- 2x - 3 d 0
(x 3)(x 1) d 0 Ž¹ÉŠÂ¥„Ážœ} 2 „¦–‡¸ °º

„¦–¸š¸É 1 ™µo ( x – 3 )(x + 1) = 0 ‹³Å—o x = 3 ®¦°º -1
„¦–š¸ ɸ 2 ™oµ ( x – 3 )( x + 1 ) < 0

Ÿ¨‡¼–…°Š‹Îµœªœš´ŠÊ ­°ŠÄœªŠÁ¨È œ°o ¥„ªnµ 0 ®¤µ¥™¹ŠÁžœ} ¨ ­—ŠªnµªŠÁ¨È ®œÉŠ¹ Áž}œ‹Îµœªœ
ª„ °¸„ªŠÁ¨È ®œŠ¹É Ážœ} ‹µÎ œªœ¨ Â¥„Å—Áo ž}œ 2 „¦–¸ ‡°º

„¦–š¸ ɸ 2.1 ™µo (x-3) > 0 ¨³ (x+1) < 0
‹³Å—o x > 3 ¨³ x<-1
Ž¹ÉŠÅ¤n¤‹¸ 圪œ‹¦·ŠÄ—Çš¸É¤µ„„ªµn 3 ¨³œ°o ¥„ªnµ –1Ĝ…–³Á—¸¥ª„œ´

„¦–¸š¸É 2.2 ™oµ (x-3) < 0 ¨³ (x+1) > 0
‹³Å—o x < 3 ¨³ x > -1
‡nµ x Ž¹ÉŠ œo°¥„ªnµ 3 ¨³¤µ„„ªnµ –1 Ĝ…–³Á—¸¥ª„´œ‡º° -1 < x < 3

Á…¥¸ œÂ­—ŠÃ—¥ÄÁo ­oœ‹µÎ œªœÅ——o ´Šœ¸Ê

72 ⌫ ⌫  ⌦
 ⌫     ⌫  

-1 3
‹µ„„¦–¸ 2.1 ¨³ 2.2‹³Å—o‡µÎ ˜°…°Š„¦–š¸ ɸ 2 ‡°º -1 < x < 3

—Š´ œ´œÊ ‡µn …°Š x šš¸É εĮ°o ­¤„µ¦Ážœ} ‹¦·Š‡°º -1 d x d3 (ŽÉ¹ŠÁ„·—‹µ„¥Á¼ œ¸¥œ…°Š‡µÎ ˜° „¦–¸šÉ¸ 1 ¨³ 2 )
—´Šœœ´Ê ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦ x2 2x d 3 ‡°º {x | -1 d x d3 } ®¦°º [-1,3]
4.8 ‡¦¼¥Êε„´œ´„Á¦¸¥œ°¸„‡¦Ê´Šªnµ Ĝ„µ¦Â„o°­¤„µ¦ ™oµÄœÃ‹š¥r°­¤„µ¦Áž}œÁ‡¦Éº°Š®¤µ¥

t ®¦°º d Ä®œo ´„Á¦¸¥œ¡·‹µ¦–µ „¦–¸ “ = ” „n°œÁ­¤° ¨oª‹¹Š¡·‹µ¦–µ„¦–¸ > (¤µ„„ªnµ ) ®¦º°
< (œo°¥„ªnµ) Á¤Éº°œ´„Á¦¸¥œš»„‡œÁ…oµÄ‹˜¦Š„´œÂ¨oª Ä®ošÎµÂ f„®´— 2.5 … …o° 5-14 Ĝ®œ´Š­º°
­µ¦³„µ¦Á¦¸¥œ¦Áo¼ ¡¤É· Á˜¤· ‡–·˜«µ­˜¦r Á¨n¤ 1 …°Š­­ªš.

5. ®¨Šn „µ¦Á¦¸¥œ¦¼o
5.1 Á°„­µ¦ f„®—´ š¸É 9
5.2 ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦Á¼o ¡·¤É Á˜·¤ ‡–·˜«µ­˜¦r Á¨¤n 1 Ê´œ¤´›¥¤«¹„¬µžše ¸É 4

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤·œŸ¨

1. ­´ŠÁ„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡µÎ ™µ¤Å—™o ¼„˜°o ŠÁž}œ­ªn œ¤µ„

2. šÎµÁ°„­µ¦ f„®—´ šÉ¸ 9 2. œ´„Á¦¥¸ œšµÎ ŗo™„¼ ˜o°Šž¦³¤µ– 85 %

3. šÎµÃ‹š¥r  f„®—´ 2.5 … …°o 5 – 14 3. œ„´ Á¦¥¸ œšµÎ ŗo™„¼ ˜°o Šž¦³¤µ– 80 %

Ĝ®œŠ´ ­°º Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡·É¤Á˜¤·

‡–˜· «µ­˜¦r

7. ´œš„¹ ®¨Š´ ­°œ
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8. „‹· „¦¦¤Á­œ°Âœ³
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⌦ 73
⌦

Á°„­µ¦ „f ®—´ šÉ¸ 9

‹ŠÂ„o°­¤„µ¦˜n°Åžœ¸Ê
1. x-2 3
2. 2x+3 d 7
3. 2x+4 d 11
4. 3-2x d 7
5. 3x+5 t 8
6. 7-8x t 6
7. –10 d 2x+4 d 6

8. 9-2x 4x-3 d x+6

74 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 75
⌦

­¦oµŠ˜µ¦µŠ x 1 1 x 3 x!3
_ + +
x-1 _ _ +
x-3 + _ +
(x-1)(x-3)

‹³Á®œÈ ŗªo nµ nªŠšŸ¸É ¨‡–¼ …°Š (x-1) ¨³ (x-3) Ážœ} +
®¦°º (x-1)(x-3) > 0 ‡º° x <1 ®¦°º x >3

—Š´ œœ´Ê ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡º° { x | x < 1®¦°º x >3 } = (- f , 1) ‰( 3,f)
®¤µ¥Á®˜» ™oµ (x-1)(x-3)d 0

‹³Á®ÈœÅ—oªnµ nªŠš¸ÉŸ¨‡¼–…°Š (x-1) ¨³ (x-3) Áž}œ ¨®¦º°«¼œ¥r‡º° 1< x <3 ®¦º° x ¤¸‡nµ
Áž}œ 1 ®¦°º 3

—´Šœ´œÊ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡°º { x | 1 d x d 3} ®¦°º [ 1, 3 ]

˜ª´ °¥nµŠš¸É 6 ‹Š®µÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦ (x + 1)(x-2)(x+3) < 0

‹µ„Ëš¥r ™µo (x+1) (x -2) (x + 3) = 0

‹³Å—o x = -1, 2 ,-3

¡·‹µ¦–µœÁ­oœ‹Îµœªœ x <-3 Ñ -3 <x <-1 Ñ -1 < x < 2 Ñ x > 2

-3 -1 2

­¦µo Š˜µ¦µŠ

x <-3 -3 <x <-1 -1 < x < 2 x > 2

x+ 3 _ + + +

x+1 _ _ + +

x–2 _ _ _ +

(x+3)(x+1)(x-2) _ + _ +

‹µ„˜µ¦µŠ nªŠšŸÉ¸ ¨‡–¼ …°ŠšŠ´Ê 3 ªŠÁ¨È Ážœ} ¨ ‡°º x < -3 ®¦º° -1 < x < 2
œœÉ´ ‡º° (x + 1)(x-2)(x+3) < 0 Á¤ºÉ° x<-3 ®¦º° -1 < x < 2
—´ŠœÊ´œÁŽ˜‡Îµ˜°…°Š°­¤„µ¦‡°º {x | x<-3 ®¦º° -1 < x < 2 }
®¦°º (f , -3) ‰ (-1 , 2 )

76 ⌫ ⌫  ⌦
 ⌫     ⌫  

4. „¦³ªœ„µ¦‹—´ „µ¦Á¦¥¸ œ¦¼o
4.1 °›·µ¥Â f„®´— 2.5 … …o° 5 – 14 ×¥ÁŒ¡µ³…o°šÉ¸œ´„Á¦¸¥œ­nªœÄ®nšÎµÅ¤nŗo (Ž¹ÉŠ¡

Á®œÈ ‹µ„„µ¦˜¦ª‹Â f„®´—) ¨³…o°šÉ¸œ„´ Á¦¸¥œ­Š­´¥ Á¡ºÉ°Áž}œ„µ¦ššªœ„µ¦Â„°o ­¤„µ¦Ã—¥¡·‹µ¦–µ
Áž}œ„¦–¸

4.2 ‡¦¼°„œ´„Á¦¸¥œªnµ Ĝ„µ¦®µ‡Îµ˜°Ã—¥Â¥„Áž}œ„¦–¸šÉ¸Å—o„¨nµª¤µÂ¨oª ™oµ¡®»œµ¤š¸É˚¥r
„ε®œ—Ä®Äo œ°­¤„µ¦¤—¸ ¸„¦­¸ ¼Š„ªµn 2 „µ¦¡·‹µ¦–µ‹³¤¸®¨µ¥„¦–¸ ŽÉŠ¹ ‹³Á­¸¥Áª¨µ¤µ„Äœ„µ¦Â„o°­¤„µ¦
—´Šœ´ÊœÁ¦µ‹³Äoª›· ¸¡‹· µ¦–µœÁ­oœ‹µÎ œªœ‹³­³—ª„„ªnµ ŽÉ¹Š‹³Äoŗo˜´ŠÊ ˜—n „¸ ¦­¸ °ŠÁžœ} ˜œo Ş

4.3 ‡¦¼¥„˜´ª°¥nµŠš¸É 5 ¨³ 6 ¨oªÄoª·›¸„µ¦™µ¤ Ä®oœ´„Á¦¸¥œ˜°š»„…Ê´œ˜°œ Á¤Éº°¤¸œ´„Á¦¸¥œµŠ
‡œ¥Š´ ŤnÁ…µo ċ­ªn œÅ®œ ‡¦‹¼ ³°›· µ¥…Ê´œ˜°œ°¥nµŠ¨³Á°¸¥—°„¸ ‡¦Ê´Š ¨ªo ‡¦¼¥„˜ª´ °¥nµŠÃ‹š¥°r „¸ 1 …°o ­¤n»
Ä®œo „´ Á¦¥¸ œªn ¥„œ´ °°„¤µšÎµœ„¦³—µœ ¨oª‡¦¼Â¨³Á¡É°º œÇnª¥„´œ˜¦ª‹­°‡ªµ¤™„¼ ˜°o Š

4.4 ‡¦¼¸Êœ³Ä®œo ´„Á¦¸¥œ­´ŠÁ„˜ªnµ Ĝ„µ¦¡·‹µ¦–µœÁ­oœ‹ÎµœªœÂ¨³œÎµÅž­¦oµŠ˜µ¦µŠš»„‡¦Ê´Š
­´¤ž¦³­·š›·Í…°Š x ˜o°ŠÁž}œª„Á­¤° ¨³Äœ„µ¦Á…¸¥œÂ˜n¨³¦¦š´—…°Š˜µ¦µŠ ˜o°ŠÁ¦¸¥Š¨Îµ—´‹µ„
ªŠÁ¨ÈšÉ¸šÎµÄ®oŗo‡nµª·„§˜œo°¥š¸É­»—¨Š¤µÁ­¤° ‹³šÎµÄ®o„µ¦¡·‹µ¦–µÁ‡¦ºÉ°Š®¤µ¥­³—ª„¨³Å¤n
Ÿ·—¡¨µ—

4.5 Ä®œo „´ Á¦¥¸ œšµÎ  „f ®´— 2.5 … …°o šÉÁ¸ ®¨°º Ĝ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¸¥œ¦oÁ¼ ¡¤É· Á˜·¤
‡–˜· «µ­˜¦r Á¨¤n 1 ´Êœ¤›´ ¥¤«¹„¬µžše ¸É 4

4.6 ‡¦¼ÁŒ¨¥Â f„®´— 2.5 ÁŒ¡µ³…o°š¸Éœ´„Á¦¸¥œ­Š­´¥ ¨³Ä®oœ´„Á¦¸¥œ­´ŠÁ„˜˜µ¦µŠ ‹µ„
˜´ª°¥nµŠ 5 ¨³ 6 ¨³‹µ„ „f ®—´ šÉ¸œ„´ Á¦¥¸ œšÎµ œ„´ Á¦¸¥œ‡ª¦‹³­Š´ Á„˜Á®ÈœªnµÁ‡¦Éº°Š®¤µ¥ Ĝ¦¦š´—
­—» šµo ¥…°Š˜µ¦µŠ…ªµ¤º°­»—‹³Ážœ} + Á­¤° ¨³™—´ ¤µšµŠŽµo ¥‹³Áž}œ – ¨oªÁžœ} + ­¨´ „œ´ š»„…o°

4.7 ‡¦¼°„œ´„Á¦¸¥œªnµ Ĝ„¦–¸š´ÉªÅž ™oµ (x-a)(x-b)(x-c) = 0 ŽÉ¹Š a < b < c ‹³Å—o‡nµ x = a
®¦º° b ®¦º° c œÁ­œo ‹Îµœªœ—´ŠœÊ¸

™oµ (x – a)( x – b)( x – c ) > 0 ‡µÎ ˜°‡°º ¥Á¼ œ¥¸ œ…°Šªn ŠšÉ¸Áž}œ +

™oµ (x – a)( x – b)( x – c ) < 0 ‡Îµ˜°‡º°¥¼Áœ¥¸ œ…°Šªn Šš¸ÉÁžœ} –

Áœn ( x – 1 )( x + 3 )( x – 5 ) > 0 ¡‹· µ¦–µ‡µn x =1, -3, 5 ¡·‹µ¦–µœÁ­oœ‹µÎ œªœ

x < -3 -3< x <1 1< x <5 x >5

- -3 + 1 - 5 +

‡Îµ˜°…°Š°­¤„µ¦‡°º -3 < x < 1 ®¦°º x > 5
ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡º° (-3,1) ‰ (5, f)

⌦ 77
⌦

4.8 ‡¦¼Ä®o˜´ª°¥nµŠÁ¡·É¤Á˜·¤š¸ÉÁž}œÁ«¬­nªœ…°Š¡®»œµ¤ ץnŠ„¨n»¤ „¨»n¤¨³ 4 ‡œÄ®oœ´„Á¦¸¥œ
nª¥„œ´ «„¹ ¬µ‹µ„Á°„­µ¦Âœ³ÂœªšµŠš¸É 10

4.9 ‡¦¼ÁŒ¨¥Á°„­µ¦Âœ³ÂœªšµŠšÉ¸ 10 ×¥„µ¦™µ¤Ä®œo „´ Á¦¸¥œªn ¥„´œ˜°
4.10 ‹„Á°„­µ¦‡ªµ¤¦o¼šÉ¸ 10 Ä®oœ´„Á¦¸¥œš»„‡œÄœ„¨»n¤nª¥«¹„¬µ ™oµÄ‡¦¤¸ž{®µŽ´„™µ¤
‡¦¼‹³°›· µ¥¥„˜´ª°¥µn ŠÁ¡¤·É Á˜·¤Â¨ªo Ä®onª¥„´œ˜°
4.11 ‡¦¼Äoª·›¸„µ¦™µ¤˜°Á¡ºÉ°Ä®oœ´„Á¦¸¥œ­¦»žÄ®oŗoªnµ Ĝ„µ¦Â„o°­¤„µ¦ ˜o°ŠšÎµ˜µ¤…´Êœ˜°œ
—´Š˜n°Åžœ¸Ê

1. šµÎ Ä®…o µo Š…ªµ…°Š°­¤„µ¦Áž}œ«œ¼ ¥Ár ­¤°
2. Á…¸¥œÄ®°o ¥Än¼ œ¦ž¼ ˜ª´ ž¦³„°…°Š¡®»œµ¤®¦º°Ÿ¨‡–¼ …°Š¡®œ» µ¤—„¸ ¦®¸ œŠÉ¹
3. ¡‹· µ¦–µœÁ­œo ‹µÎ œªœ
4. ‡Îµ˜°…°Š°­¤„µ¦˜°Äœ¦ž¼ ÁŽ˜®¦°º nªŠ
4.12 Â¸Ê ‹Š„µ¦š—­°¥n°¥Á„ȝ‡³ÂœœÁ¦°Éº Š„µ¦Â„°o ­¤„µ¦Äœ‡µ˜n°Åž

5. ®¨nŠ„µ¦Á¦¥¸ œ¦o¼
5.1 Á°„­µ¦Âœ³ÂœªšµŠš¸É 10
5.2 Á°„­µ¦‡ªµ¤¦š¼o ¸É 10
5.3 ®œŠ´ ­°º Á¦¸¥œ­µ¦³„µ¦Á¦¥¸ œ¦¼oÁ¡·¤É Á˜·¤ ‡–·˜«µ­˜¦r Á¨n¤ 1 ´Êœ¤´›¥¤«„¹ ¬µžešÉ¸ 4

6. „¦³ªœ„µ¦ª—´ ¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤œ· Ÿ¨

1. ­Š´ Á„˜‹µ„„µ¦˜°‡Îµ™µ¤ 1. œ´„Á¦¥¸ œ˜°‡µÎ ™µ¤Å—o™„¼ ˜o°Š Áž}œ­ªn œ¤µ„

2. ­Š´ Á„˜‹µ„„µ¦¦nª¤„·‹„¦¦¤ 2. œ´„Á¦¥¸ œ¦ªn ¤„‹· „¦¦¤—¸

3. ­Š´ Á„˜‹µ„‡ªµ¤­œÄ‹ 3. œ„´ Á¦¥¸ œÄ®o‡ªµ¤­œÄ‹—¸ ¨³˜ÊŠ´ ċÁ¦¥¸ œ

4. šµÎ ˚¥Ár °„­µ¦Âœ³ÂœªšµŠšÉ¸ 10 4. œ´„Á¦¸¥œšÎµ™„¼ ˜o°Š 95 %

5. šµÎ  f„®—´ 2.5 … Ĝ®œ´Š­°º Á¦¥¸ œ­µ¦³ 5. œ´„Á¦¥¸ œšÎµÅ—™o „¼ ˜°o Šž¦³¤µ– 85 %

„µ¦Á¦¸¥œ¦oÁ¼ ¡·¤É Á˜¤· ‡–·˜«µ­˜¦r Á¨n¤ 1

7. ´œš„¹ ®¨´Š­°œ
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8. „‹· „¦¦¤Á­œ°Âœ³
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78 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦Âœ³ÂœªšµŠš¸É 10

‡Îµ­Š´É Ä®oœ„´ Á¦¸¥œ°µn œšµÎ ‡ªµ¤Á…µo ċ ¨ªo Á˜¤· ‡Îµ˜°¨ŠÄœ°n ŠªnµŠ

1. ‹ŠÂ„°o ­¤„µ¦ x 4 t 0 4 ¨³ x z ………. 4
-3
x 3

™µo x 4 0 ‹³Å—o x

x 3

čo x = 4

­¦oµŠ˜µ¦µŠ

x -3 -3 x 4 x!4

x 3

x 4
x 4
x 3

‹µ„˜µ¦µŠ x 4 ! 0 Á¤É°º …..………. ®¦º° …..……….

x 3

—Š´ œÊ´œ x 4 t 0 Á¤°Éº …..………. ®¦°º …..……….

x 3

—´Šœ´ÊœÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡°º …..……….…..……….

2. ‹ŠÂ„o°­¤„µ¦ x 1 x 2 d 0
x 4

™µo x 1 x 2 0 ‹³Å—o x …..………. ®¦°º …..……….
x 4

˜n x z …..……….

č‡o nµ x …..……….

­¦oµŠ˜µ¦µŠ

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⌦ 79
⌦

3. ‹ŠÂ„o°­¤„µ¦ 4 t 1

x 2 x 1

ª›· ¸šÎµ 4 1 t 0

x 2 x 1

(Ä®oœ´„Á¦¥¸ œšÎµÄ®oÁžœ} Ÿ¨­ÎµÁ¦‹È ץčo‡ªµ¤¦oÁ¼ ¦º°É Š„µ¦ª„ ¨Á«¬­nªœ…°Š¡®œ» µ¤)
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80 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦‡ªµ¤¦oš¼ ¸É 10

˜ª´ °¥nµŠšÉ¸ 1 ‹ŠÂ„°o ­¤„µ¦ x2 2x 1 t 0
ª›· ¸šµÎ ‹µ„ x2 2x 1 t 0
‹³Å—o (x 1)2 t 0
(x 1)2 t 0 Á­¤°Å¤nªnµ x ‹³Áž}œ‹µÎ œªœ‹¦Š· Ä—Ç „˜È µ¤
—´Šœ´œÊ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡°º R

˜ª´ °¥µn ŠšÉ¸ 2 ‹ŠÂ„°o ­¤„µ¦ x2 2x 3 ! 0

ª·›š¸ µÎ ‹µ„ x2 2x 3 ! 0

x2 2x 1 2 ! 0

( x 1)2 2 ! 0

(x 1)2 t 0 Á­¤°Å¤nªµn x ‹³Áž}œ‹µÎ œªœ‹¦·ŠÄ—Ç „Șµ¤

—´ŠœœÊ´ (x 1)2 2 t 0 ­Îµ®¦´ š»„‡nµ xR

ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡°º {x| xR }

˜´ª°¥µn ŠšÉ¸ 3 ‹ŠÂ„°o ­¤„µ¦ x2 2x 3 0

ª›· ¸šÎµ x2 2x 3 0

x2 2x 1 2 0

x 1 2 2 0
(x 1)2 t 0 Á­¤°Å¤ªn nµ x ‹³Ážœ} ‹µÎ œªœ‹¦·ŠÄ—Ç „Șµ¤

—Š´ œÊ´œ x 1 2 2 0 ‹Š¹ Ť¤n ‡¸ nµ x šÉ¸šµÎ Ä®o°­¤„µ¦Áž}œ‹¦·Š

ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡º°I

˜´ª°¥µn ŠšÉ¸ 4 ‹ŠÂ„°o ­¤„µ¦ x2 4x 4 ! 0
ª·›š¸ ε x2 4x 4 ! 0
x 2 2 ! 0
x 2 2 ! 0 Á­¤°Å¤nªnµ x ‹³Ážœ} ‹µÎ œªœ‹¦·ŠÄ—Ç „Șµ¤

˜n x 2 2 ! 0 Á¤É°º x z 2 —Š´ œ´œÊ ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦ ‡°º R - {2}

⌦ 81
⌦

˜´ª°¥nµŠšÉ¸ 5 ‹ŠÂ„o°­¤„µ¦ x2 4x 4 d 0
ª·›¸šµÎ x2 4x 4 d 0
x 2 2 d 0
ÁœºÉ°Š‹µ„ x 2 2 t 0 Á­¤°Å¤ªn µn x ‹³Áž}œ‹Îµœªœ‹¦Š· Ä—Ç „Șµ¤
˜n x 2 2 0 ‡µÎ ˜°‡°º I , x 2 2 0 Á¤ºÉ° x 2
—Š´ œÊ´œ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡°º {2}

˜´ª°¥nµŠš¸É 6 ‹ŠÂ„o°­¤„µ¦ x2 4x 4 d 0

ª›· š¸ µÎ x2 4x 4 d 0

x2 4x 4 8 d 0

x 2 2 2

8 d0
x 2 8 x 2 8 d 0

+- +

2 8 2 8

—Š´ œœÊ´ ÁŽ˜‡Îµ˜°…°Š°­¤„µ¦ ‡º° ^x | 2 2 2 d x d 2 2 2`

…°o ˜°n ŞĮoœ´„Á¦¥¸ œšÎµÁ°Š

‹ŠÂ„o°­¤„µ¦ (x 1)3 (x 3)2 t0
(x 4)5

(Ä®œo ´„Á¦¥¸ œšµÎ Á°Š)

.......................................................................................................................................................................

.......................................................................................................................................................................

.......................................................................................................................................................................

‹ŠÂ„o°­¤„µ¦ (x 2)(3 x) 0
(Ä®œo „´ Á¦¥¸ œšµÎ Á°Š)
.......................................................................................................................................................................

.......................................................................................................................................................................

.......................................................................................................................................................................

82 ⌫ ⌫  ⌦
 ⌫     ⌫  

Ÿœ„µ¦‹´—„µ¦Á¦¸¥œ¦o¼š¸É 11

Á¦ºÉ°Š ‡µn ­¤´ ¼¦–r ´Êœ¤›´ ¥¤«„¹ ¬µžeš¸É 4
ª· µ ‡–·˜«µ­˜¦r Áª¨µ 2 ´ªÉ äŠ

***********************************************************************************

Ÿ¨„µ¦Á¦¥¸ œ¦šo¼ ‡¸É µ—®ª´Š
„­o ¤„µ¦˜´ªÂž¦Á—¥¸ ªš¸É°¥Än¼ œ¦ž¼ ‡µn ­´¤¦¼ –rŗo

1. ‹»—ž¦³­Š‡„r µ¦Á¦¥¸ œ¦¼o œ´„Á¦¥¸ œ­µ¤µ¦™
1.1 °„‡ªµ¤®¤µ¥Â¨³­¤˜´ ·…°Š‡µn ­´¤¦¼ –rŗo
1.2 „o­¤„µ¦˜´ªÂž¦Á—¸¥ªš°¸É ¥¼Än œ¦¼ž‡µn ­´¤¼¦–rŗo

2. œª‡ªµ¤‡—· ®¨´„
‡nµ­´¤¼¦–rĜÁ·ŠÁ¦…µ‡–·˜ Áž}œ¦³¥³®nµŠ¦³®ªnµŠ‹»—šœ 0 „´‹»—šœ a Á¤ºÉ° a Áž}œ‹Îµœªœ‹¦·Š

œÁ­œo ‹µÎ œªœ Ĝ¦³‹Îµœªœ‹¦·Š ‡nµ­´¤¦¼ –…r °Š‹Îµœªœ‹¦·Š a ‹³Âšœ—oª¥­´ ¨„´ ¬–r |a|

°°­a Á¤ºÉ° a ! 0
®0 Á¤°Éº a 0
×¥šÉ¸ a

°
°¯ a Á¤É°º a 0

‡ªµ¤¦o¼Â¨³‡ªµ¤Á…oµÄ‹Á¦Éº°Š‡nµ­´¤¼¦–r ‹³Áž}œž¦³Ã¥œrĜ„µ¦Â„o­¤„µ¦Â¨³°­¤„µ¦šÉ¸

Á„¥¸É ª„´‡nµ­¤´ ¼¦–r¨³Á¦º°É Š°ºÉœÇ ˜n°Åž

3. ÁœÊ°º ®µ­µ¦³

3.1 œ·¥µ¤ Á¤°Éº a Ážœ} ‹µÎ œªœ‹¦·ŠÄ—Ç ‡µn ­´¤¦¼ –…r °Š‹Îµœªœ‹¦·Š a šœ—ªo ¥ |a|

°­a Á¤É°º a t 0
a °¯® a Á¤ºÉ°a 0

3.2 š§¬‘¸š Á¤°Éº x ¨³ y Ážœ} ‹Îµœªœ‹¦Š·

1. x x 4. x y y x

2. xy x y 5. x 2 x 2

3. x x , y z 0 6. x y d x y

yy

⌦ 83
⌦

4. „¦³ªœ„µ¦‹´—„µ¦Á¦¸¥œ¦o¼
4.1 š—­°Á„ȝ‡³ÂœœÁ¦É°º Š„µ¦Â„o°­¤„µ¦ 20 œµš¸
4.2 ‡¦¼ššªœ„µ¦Á…¸¥œÁ­oœ‹ÎµœªœÂ¨³‡nµ­´¤¼¦–rš¸ÉÁ‡¥Á¦¸¥œ¤µÂ¨oª ĜÊ´œ¤´›¥¤«¹„¬µžeš¸É 2

‡¦Á¼ …¸¥œÁ­oœ‹Îµœªœ—Š´ ¦ž¼

-3 0 3

‹µ„Á­oœ‹Îµœªœ ‡¦¼™µ¤œ´„Á¦¸¥œªµn ¦³¥³šµŠ‹µ„ 3 ™¹Š 0 „´¦³¥³šµŠ‹µ„ –3 ™¹Š 0 Ášnµ„´œ®¦º°Å¤n ( Ášnµ„´œ
Á¡¦µ³°¥¼n®µn Š‹µ„‹—» 0 Áž}œ¦³¥³ 3 ®œnª¥Ášµn „œ´ )

‡¦¼°„œ´„Á¦¸¥œªnµ¦³¥³šµŠ‹µ„‹»— 0 ™¹Š‹»— 3 œÁ­oœ‹ÎµœªœÁ¦¸¥„ªnµ‡nµ­´¤¼¦–r…°Š 3
Á…¸¥œÂšœ—ªo ¥ 3

‡¦¼°„œ´„Á¦¸¥œªnµ¦³¥³šµŠ‹µ„‹»— 0 ™¹Š ‹»— –3 œÁ­oœ‹ÎµœªœÁ¦¸¥„ªnµ‡nµ­´¤¼¦–r…°Š –3
Á…¥¸ œÂšœ—ªo ¥ 3

‡¦¼™µ¤œ´„Á¦¸¥œªnµ 3 „´ 3 Ášnµ„´œ®¦º°Å¤nÁ¡¦µ³Á®˜»Ä— (Ášnµ„´œ Á¡¦µ³°¥¼n®nµŠ‹µ„‹»— 0 Áž}œ
¦³¥³ 3 ®œªn ¥Ášnµ„œ´ )

4.3 ‡¦¼˜´ŠÊ ‡Îµ™µ¤Á„ɸ¥ª„´ ‡nµ­´¤¦¼ –…r °Š‹ÎµœªœÂ˜¨n ³‹Îµœªœ ×¥­¤»n œ´„Á¦¥¸ œÄ®˜o °š¨¸ ³‡œ
¡¦°o ¤š´ÊŠÄ­o ´ ¨´„¬–Âr šœ‡µn ­¤´ ¼¦–Är ®™o „¼ ˜o°Š Ánœ

‹—» °³Å¦š°É¸ ¥®n¼ µn Š‹µ„‹»— 0 Áž}œ¦³¥³ 8 ®œnª¥ œ„´ Á¦¸¥œ‡ª¦˜°Å—ªo µn 8 ¨³ –8 ­¦»žÅ—ªo nµ‡nµ
­¤´ ¼¦–r…°Š 8 ¨³ –8 ‡°º 8 Á…¥¸ œÂšœ—ªo ¥ 8 8 ®¦°º 8 8

4.4 ‡¦¼˜´ÊŠ‡Îµ™µ¤Ã—¥„ε®œ—‡nµ­´¤¼¦–r…°Š‹Îµœªœ‹¦·Š ¨oªÄ®oœ´„Á¦¸¥œ°„‡ªµ¤®¤µ¥Â¨³‡nµ
­¤´ ¼¦–…r °Š‹Îµœªœ‹¦Š· œœÊ´ Ç Ã—¥¥„˜ª´ °¥nµŠœÎµ Ánœ

2 Áž}œ¦³¥³šµŠ‹µ„ 0 ™¹Š‹»—šœ‹µÎ œªœ 2 ŽŠÉ¹ Ášnµ„´ 2 ®œnª¥ —´Šœ´œÊ 2 2
|-1| Áž}œ¦³¥³šµŠ‹µ„ 0 ™¹Š…………………………………… —Š´ œœÊ´ |-1| = …..
| -2| …………………...……………………………………… —Š´ œÊ´œ |-2 | = .….
0 ………………….……..……………………………….... —´Šœœ´Ê | 0 | = ….

™µo a Áž}œ‹µÎ œªœª„ a a

™µo a Áž}œ«¼œ¥r 00

™µo a Áž}œ‹µÎ œªœ¨ a a Áœn a = -3 ‹³Å—o |a| = | -3 | = - (-3) = 3

a = -5 ‹³Å—o | a | = | -5 | = -(-5) = 5

4.5 ‡¦Ä¼ ®œo „´ Á¦¥¸ œªn ¥„œ´ ­¦»žÁž}œœ¥· µ¤…°Š‡µn ­¤´ ¼¦–r Á¤É°º a Ážœ} ‹µÎ œªœ‹¦·ŠÄ—Ç œ„´ Á¦¸¥œ‡ª¦

­¦»žÅ—oªnµ

84 ⌫ ⌫  ⌦
 ⌫     ⌫  

a °­°a Á¤Éº° a Ážœ} ‹µÎ œªœª„
®0 Á¤°Éº a 0

°
°¯ a Á¤°ºÉ a Áž}œ‹Îµœªœ¨

4.6 Ä®oœ´„Á¦¸¥œšÎµÁ°„­µ¦Âœ³ÂœªšµŠš¸É 11.1 Á¡ºÉ°­¦»ž­¤´˜·µŠž¦³„µ¦…°Š‡nµ­´¤¼¦–r

ŽÉŠ¹ œ„´ Á¦¥¸ œ‡ª¦­¦»žÅ——o ´ŠœÊ¸

4.6.1 x x

4.6.2 xy x y

x x ,y z0

4.6.3
yy

4.6.4 x y y x

4.6.5 x 2 x 2

4.6.6 x y d x y

4.6.7 x y t x y

4.7 Ä®oœ„´ Á¦¸¥œšµÎ Á°„­µ¦Âœ³ÂœªšµŠš¸É 11.2 Á¡°Éº ­¦»žÁžœ} šœ¥· µ¤ ŽÉ¹Šœ´„Á¦¸¥œ‡ª¦­¦»žÅ—o—Š´ œÊ¸

šœ¥· µ¤ Á¤º°É x Áž}œ‹Îµœªœ‹¦·Š ¨³ a > 0, x a ®¤µ¥™¹Š x a ®¦°º x a

4.8 ®¨´Š‹µ„­¦»žœ·¥µ¤Â¨³š»„‡œÁ…oµÄ‹˜¦Š„´œÂ¨oª ‡¦¼Ä®o˜´ª°¥nµŠÁ¡·É¤Á˜·¤Ä®oœ´„Á¦¸¥œšÎµ
 f„®´— 2.6 …o° 1-3 Ĝ®œ´Š­º°Á¦¸¥œ­µ¦³„µ¦Á¦¸¥œ¦¼oÁ¡É·¤Á˜·¤ ¨³Ä®ošÎµÁ°„­µ¦ f„®´— 11.1 Áž}œ
„µ¦oµœ

4.9 ‡¦Á¼ Œ¨¥Á°„­µ¦ f„®—´ š¸É 11.1 ¨³°›·µ¥…o°š¸Éœ´„Á¦¸¥œ­Š­´¥ ¡¦o°¤„´¥ÎʵªnµÄœ„µ¦Â„o­¤„µ¦
˜o°Š˜¦ª‹‡µÎ ˜°š»„‡¦Š´Ê

4.10 ‡¦¼ššªœ­¤´˜·…°Š‡nµ­´¤¼¦–rÁ„ɸ¥ª„´ x 2 x2 ¨³ x x Ž¹ÉŠ‹³œÎµÅžÄoĜ„µ¦

yy

„o­¤„µ¦‡nµ­´¤¼¦–r
4.11 ‡¦¼¥„˜´ª°¥nµŠÃ‹š¥r ¨oªÄ®oœ´„Á¦¸¥œnª¥„´œšÎµ ×¥‡¦¼™µ¤Ä®oœ´„Á¦¸¥œ˜° Á¡ºÉ°Ä®oœ´„Á¦¸¥œ

­µ¤µ¦™­¦»žª·›š¸ µÎ ŗo

⌦ 85
⌦

˜ª´ °¥nµŠš¸É 1 ‹ŠÂ„­o ¤„µ¦ 2 x x 3
ª›· š¸ µÎ ‹µ„ 2 x x 3
¥„„ε¨Š´ 2; 2 x 2 x 3 2

4 – 4x + x2 = x2 + 6x + 9

- 10x = 5

ÁŽ˜‡Îµ˜°‡º° x 1
2

­®¯ 21 ¿¾½

˜ª´ °¥nµŠš¸É 2 ‹ŠÂ„­o ¤„µ¦ x 2 2

x 1

ª›· š¸ µÎ ‹µ„ xx
yy

—Š´ œÊœ´ x 2 2

x 1

x 2 2x 1

x 2 2 2x 2 2

x 2 2 2x 2 2

x 2 2 2x 2 2 0

> x 2 2x 2 @> x 2 2x 2 @ 0

3x 4 x 0
œÉ´œ‡º° x 4 x 0

‹³Å—o x 0 ®¦º° x 4

ÁŽ˜‡µÎ ˜°‡º° ^0,4`

˜ª´ °¥µn ŠšÉ¸ 3 ‹ŠÂ„­o ¤„µ¦ 3x 2 1 2x

ª·›¸šµÎ ‹µ„ 3x 2 1 2x

‹³Å—o 3x + 2 = 1 – 2x ®¦º° 3x + 2 = - (1 – 2x)
5x = -1 ®¦º° x = -3

x = 1

5

‹µ„„µ¦˜¦ª‹‡µÎ ˜°‹³Å—o

x = 1 ¨³ x = -3

5

? ÁŽ˜‡µÎ ˜° ‡°º ­®¯ 1 , 3¾½¿
5

86 ⌫ ⌫  ⌦
 ⌫     ⌫  

4.12 Ä®oœ´„Á¦¸¥œšÎµÃ‹š¥rÁ°„­µ¦ f„®´—šÉ¸ 11.2 Á­¦È‹Â¨oªnª¥„´œÁŒ¨¥‡Îµ˜° ¨³°›·µ¥…o°
­Š­´¥ ¨³¥„˜´ª°¥µn ŠÁ¡É·¤Á˜¤· ­Îµ®¦´ œ´„Á¦¸¥œš¤É¸ ž¸ {®µ®¦°º šÎµÅ¤n™„¼

4.13 ‡¦¼ššªœœ·¥µ¤…°Š‡nµ­´¤¼¦–r¨³­¤´˜·šÉ¸­Îµ‡´…°Š‡nµ­´¤¼¦–ršÉ¸Á„¸É¥ª„´­¤„µ¦
„¨nµª‡º° |x| = a „Șn°Á¤ºÉ° x = a ®¦º° x = -a ¨oªš—­°Á„¸É¥ª„´­¤„µ¦‡nµ­´¤¼¦–r 1 …o°
( Ä®o­—Šª·›š¸ ε)

5. ®¨Šn „µ¦Á¦¸¥œ¦¼o
5.1 Á°„­µ¦Âœ³ÂœªšµŠš¸É 11.1, 11.2
5.2 Á°„­µ¦ „f ®´—šÉ¸ 11.1 , 11.2
5.3 ®œŠ´ ­º°Á¦¥¸ œ­µ¦³„µ¦Á¦¥¸ œ¦¼oÁ¡·¤É Á˜·¤ ‡–·˜«µ­˜¦r Á¨¤n 1 œÊ´ ¤´›¥¤«„¹ ¬µžše ɸ 4 …°Š ­­ªš.

6. „¦³ªœ„µ¦ª´—¨³ž¦³Á¤œ· Ÿ¨

„µ¦ª—´ Ÿ¨ „µ¦ž¦³Á¤œ· Ÿ¨

1. ­´ŠÁ„˜‹µ„„µ¦˜°‡µÎ ™µ¤ 1. œ„´ Á¦¥¸ œ˜°‡Îµ™µ¤Å—o™¼„˜°o ŠÁžœ} ­ªn œ¤µ„

2. šÎµÁ°„­µ¦Âœ³ÂœªšµŠšÉ¸ 11.1, 11.2 2. œ„´ Á¦¸¥œšÎµÅ—o™„¼ ˜o°Šž¦³¤µ– 90 %

3. šµÎ Á°„­µ¦ „f ®´—šÉ¸ 11.1, 11.2 3. œ´„Á¦¥¸ œšÎµÅ—o™¼„˜o°Šž¦³¤µ– 80 %

4. šµÎ ˚¥r f„®´— 2.6 Ĝ®œŠ´ ­º°Á¦¸¥œ 4. œ„´ Á¦¥¸ œšÎµÅ—™o „¼ ˜o°Šž¦³¤µ– 85 %

­µ¦³„µ¦Á¦¸¥œ¦o¼Á¡É·¤Á˜¤· ‡–˜· «µ­˜¦r Á¨n¤ 1 5. œ´„Á¦¸¥œÂ­—Šª·›š¸ µÎ ŗo™„¼ ˜o°Šž¦³¤µ– 80 %

5. š—­° ­—Šª·›¸šÎµ

7. œ´ š„¹ ®¨´Š­°œ
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………

8. „·‹„¦¦¤Á­œ°Âœ³
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………
……………………………………………………………………………………………………………

⌦ 87
⌦

Á°„­µ¦Âœ³ÂœªšµŠš¸É 11.1

‡µÎ ­Š´É ‹Š˜°‡Îµ™µ¤˜°n ޜʸ
1. 2 2 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° Ášµn „œ´ Á¡¦µ³˜µn ŠÁšµn „´ 2

2. 5 5 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —
˜° …………………………….

3. x x ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —
˜° …………………………….

4. 2 5 2 5 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —

˜° …………………………….

5. 2 5 2 5 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° …………………………….

6. 2 5 2 5 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —

˜° …………………………….

7. xy x y ®¦º°Å¤n Á¡¦µ³Á®˜»Ä—
˜° …………………………….

20 20 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

8.
44

˜° ……………………………

20 20 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —

9.
44

˜° …………………………….

88 ⌫ ⌫  ⌦
 ⌫     ⌫  

20 20 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —

10.
4 4

˜° …………………………….

x x ,y z0 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —

11.
yy

˜° …………………………….

12. 9 4 4 9 ®¦º°Å¤n Á¡¦µ³Á®˜»Ä—
˜° …………………………….

13. 9 4 4 9 . ®¦°º Ťn Á¡¦µ³Á®˜»Ä—

˜° …………………………….

14. 9 4 4 9 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° …………………………….

15. 9 4 4 9 ®¦°º Ťn Á¡¦µ³Á®˜»Ä—

˜° …………………………….

16. x y y x ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —
˜° …………………………….

17. 5 2 5 2 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° ……………………………….

18. 5 2 5 2 ®¦º°Å¤n Á¡¦µ³Á®˜»Ä—

˜° ……………………………….

19. x 2 x 2 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° ……………………………….

⌦ 89
⌦

20. 2 5 d 2 5 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —
˜° ……………………………….

21. 2 5 d 2 5 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —
˜° ……………………………….

22. 5 2 d 5 2 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° ……………………………….

23. 5 2 d 5 2 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° ……………………………….

24. x y d x y ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —
˜° … …………………………….

25. 5 3 t 5 3 ®¦º°Å¤n Á¡¦µ³Á®˜Ä» —
˜° ……………………………….

26. 3 5 t 3 5 ®¦º°Å¤n Á¡¦µ³Á®˜»Ä—
˜° ……………………………….

27. 5 3 t 5 3 ®¦°º Ťn Á¡¦µ³Á®˜Ä» —

˜° ……………………………….

28. x y t x y ®¦°º Ťn Á¡¦µ³Á®˜Ä» —
˜° ……………………………….

90 ⌫ ⌫  ⌦
 ⌫     ⌫  

⌦ 91
⌦

Á°„­µ¦ f„®´—šÉ¸ 11.1

Éº° …………………………………………………….œ´Ê ………………..…….Á¨…šÉ¸ ………………..
‹ŠœÎµšœ¥· µ¤Â¨³­¤´˜·…°Š‡µn ­¤´ ¼¦–¤r µÄÄo œ„µ¦Â„­o ¤„µ¦˜°n ޜʸ
1. x 3 5

2. 3 2x 3 2x

3. 3 2x 2x 3

4. x x 1

5. x x 1

6. x 2 x 3

7. x 4 8 3x

8. 2x 1 x 3

92 ⌫ ⌫  ⌦
 ⌫     ⌫  

Á°„­µ¦ „f ®—´ š¸É 11.2

ºÉ° …………………………………………………..……Ê´œ……………..……Á¨…šÉ¸ ….....……………
‡Îµ­ÉŠ´ ‹ŠÂ„­o ¤„µ¦˜n°ÅžœÊ¸
1. x 2 x 1

2. 3x 4 2x 1
3. x 3 1

x 2

4. 3x 2 2

x 1

5. 4 x 2 2 3 x 2 1 0
6. x 2 2 x 1 x 4

7. x2 4x 4 2x 4

8. x 3 x 5 2

9. x 1 x 2 x 1 5

⌦ 93
⌦

Ÿœ„µ¦‹—´ „µ¦Á¦¸¥œ¦oš¼ ¸É 12

Á¦°Éº Š ‡µn ­´¤¼¦–r ( ˜n° ) ´Êœ¤›´ ¥¤«„¹ ¬µžše ɸ 4
ª· µ ‡–˜· «µ­˜¦r Áª¨µ 2 É´ªÃ¤Š

***********************************************************************************

Ÿ¨„µ¦Á¦¥¸ œ¦š¼o ¸É‡µ—®ª´Š
„o°­¤„µ¦˜´ªÂž¦Á—¸¥ªš°¸É ¥Än¼ œ¦ž¼ ‡µn ­´¤¼¦–År —o

1. ‹»—ž¦³­Š‡r„µ¦Á¦¸¥œ¦¼o œ„´ Á¦¸¥œ­µ¤µ¦™
1.1 °„š§¬‘¸ ššÁ¸É „ɸ¥ª„´ ‡nµ­¤´ ¼¦–År —o
1.2 „°o ­¤„µ¦Äœ¦ž¼ ‡µn ­¤´ ¼¦–r˜ª´ ž¦Á—¥¸ ª°¥µn ŠŠµn ¥Å—o

2. œª‡ªµ¤‡—· ®¨´„
š§¬‘¸ š Á¤É°º a Áž}œ‹Îµœªœ‹¦·Šª„
x a ®¤µ¥™¹Š a x a
x d a ®¤µ¥™Š¹ a d x d a
x ! a ®¤µ¥™¹Š x a ®¦°º x ! a
x t a ®¤µ¥™Š¹ x d a ®¦º° x t a

3. ÁœºÊ°®µ­µ¦³

˜ª´ °¥µn ŠšÉ¸ 1 ‹Š®µ‡µn …°Š x Á¤º°É „µÎ ®œ— | 2x + 1 | < 5
ª›· ¸šÎµ | 2x+1 | < 5 ¤¸‡ªµ¤®¤µ¥Á®¤°º œ„´ -5 < 2x+1 < 5

œœ´É ‡º° -5 –1 < 2x+1-1 < 5-1

-6 < 2x < 4
—´Šœ´Êœ -3 < x < 2
ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡º° { x | -3 < x < 2 } = (-3,2)

˜´ª°¥µn Šš¸É 2 ‹Š®µ‡µn …°Š x Á¤ºÉ°„µÎ ®œ— | 4x - 2 | > 10
ª›· š¸ ε | 4x-2 | > 10 ¤¸‡ªµ¤®¤µ¥Á®¤°º œ„´
4x-2 < -10 ®¦°º 4x-2 > 10
œœÉ´ ‡º° 4x-2+2 < -10+2 ®¦º° 4x-2+2 > 10+2
4x < -8 ®¦°º 4x > 12
—´ŠœœÊ´ x < -2 ®¦º° x > 3
ÁŽ˜‡µÎ ˜°…°Š°­¤„µ¦‡º° { x | x < -2 ®¦º° x > 3 }= (f ,-2) ‰ (3, f )