TUTORIAL 2

SM015 CHAPTER 2: EQUATIONS, INEQUALITIES AND

ABSOLUTE VALUES

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TUTORIAL 2 : EQUATIONS, INEQUALITIES AND ABSOLUTE VALUES

2.1 EQUATIONS g) log3 ( x + 2) = 2 log3 x
1. Solve each of the following:
( )h) log2 (2x − 4) = 2 + l og2 x2 − 6
a) 3x+1 = 4x−1
i) 2(log4 x + logx 4) = 5
( )b) 16x − 5 22x−1 +1 = 0
j) log2 (2x+1 − 32 2) = x
c) 4x+1 − 25+x − 2x + 23 = 0
d) x + x + 7 = 7 k) 2ln x = ln 3 + ln(6 − x)

e) 2x + 5 − x −1 = 2 l) x6e−4ln x = 2x +15

f) 4x +13 − x +1 = 12 − x

2. a) By substituting a = 2x , solve the equation 4x + 3 = 2x+2 [6m]
b) Given that 81y = 3(2 y−3)x and 218y+6x = 64xy . Find the values of x and y [6m]

3. a) Solve the equation 3 log x 3 + log3 3 x = 10. [7m]
3 [6m]

b) Solve the equation 32x −10(3x−1) +1 = 0

4. a) Solve the equation ln x − 3 = −2. [6m]
ln x [6m]

b) Solve the equation 32x+1 − 28(3x ) + 9 = 0

5. a) Solve the equation log(x − 4) + 2 log3 = 1 + log x  . [5m]
[6m]
2

b) Find the value of x which satisfies the equation
log2 (5 − x) − log2 (x − 2) = 3 − log2 (1+ x)

( )( )6. a) Given log 2 = m and log7 = n . Express x in terms of m and n if 143x+1 82x+3 = 7 [6m]

b) Find the value of x which satisfies the equation log9 x = (log3 x)2 , x  1 [7m]

7. a) Solve the equation 22x−2 − 2x+1 = 2x − 23 [7m]
b) Solve the equation 3x + 3(3−x) = 12 [6m]
[6m]
8. a) Evaluate the solution of 4 y−2 = 1 up to three decimal places. [7m]
3− y [6m]
[7m]
b) Solve the equation 2 + log2 x = 15logx 2
[5m]
( )9. a) Determine the values of x which satisfy the equation 32x−1 = 4 3x − 9 [6m]
[7m]
b) Solve the equation log2 x − log4 (3x + 4) = 0

2.2 INEQUALITIES

10. Find the ranges of values of x that satisfy the following inequalities:

a) 2 − 3 ≤ 3 + d) 5 x 2 > 3 x + 2
b) −5 < 3 + 2 < −2 e) (x – 1 )2 > 9

c) x + 1  3x +1
23 4

11. Find the range of values of x for which b) x −1  x2 + 3x  x + 3
a) 3 x + 4 < x 2 – 6  9 – 2 x

12. Determine the solution set for each of the following:

a) 1  1 c) x + 3  5
1− x 4− x x−2

b) x  x + 2
x −1 x +1

13. Find the solution set of the inequality
a) 1  1
3− 2x x + 4
b) x  1
x + 4 2x −1

14. a) Determine the solution set for 2x + 3  5
x

b) Solve the following inequalities 3x2 + x − 4  0. [4m]
2x2 − 3x − 2

15. Determine the solution set of the inequality [6m]
a) 1  1 [7m]
2x −1 x + 2
b) 2x2 + 9x − 4  4
x+2

16. Solve the inequality

a) 1  1 [6m]
6− x x −1 [7m]

b) 2 −  x + 2   5
 x − 4 

2x  42x 2x  42x
8x 8x
( )17. Show that [9m]
= 22x . Hence, find the interval for x so that −13 2x + 36  0

2.3 ABSOLUTE VALUES
18. Solve the given equations by using definition.

a) x +1 = 2 e) x − 2 = 10 − 3x
x−3 f) x + 2x = 3
g) 2 x −1 = x + 4
b) 12x +1 = 3
24

c) x2 − 7 = 2

d) 4 − 3x = 5x + 4

Write down the answer in solution set form.

19. Solve [6m]
[6m]
a) x2 − x − 3 = 3 [5m]
b) 6x2 + x −11 = 4

c) 3 = 7, x  4
x−4

20. Find the solution set of each of the following:

a) 3x + 4  5 e) x −1  2x − 3

b) x + 5  4x + 8 f) 5 2x − 3  4 x − 5

c) 2x +1  3x + 2 g) x +1  2
3x − 2

d) x  2
x+4

21. By completing the square express the inequality x2 + 8x + c  0 in the form x + a  b where a
and b are constants. Therefore, find the interval of x so that x2 + 8x +10 is always more than 3.

22. a) Obtain the solution set for 2x +1  −x2 + 4 [7m]
b) Determine the interval of x satisfying the inequality x + 2  10 − x2 [7m]

23. a) Solve 2 5 + x  x [4m]
b) Find the solution set of 2 − 3x  x + 3 [8m]

24. Solve the following inequality equation for all x is real numbers. Write your answer in set form.

4 − 3− 2x 1 [7m]
1+ x

25. Solve the following inequalities [7m]
a) x  2 [8m]
x+4
b) x −1  2
x+3

26. Solve the equation 32x+1 − (16)3x + 5 = 0 [6m]
[6m]
27. a) Solve 6x +1 − x = 3

b) Determine the solution set of x which satisfy the inequality 2  x [7m]
x +1 x +3
[4m]
28. Find the interval of x for which the following inequalities are true [9m]
a) 5 −1  0
x+3 [5m]
b) 3x − 2  2 [5m]
2x +3
[6m]
29. Solve the following: [7m]
[6m]
( )a) 3 52x 1 x+1 [5m]
[8m]
+ 252 = 200

b) x + 4  x2 + x  12

30. Solve the following:

a) log2 2x = 2 log4 ( x + 4)

b) 2 x − 3  1
2x −1

31. If 7 − 3 5 = x − y , determine the values of x and y
32. a) Solve the following equation 3 = 7, x  4

x−4
b) Find the solution set for the inequality −4 − x  x + 4, x  3

x−3

Answer:

1. a) x = 8.638 b) x = − 1 , 1 c) x = −2,3
22 f) x = 3
d) x = 9
e) x = 2,10

g) x = 2 h) x = 5 i) x = 2,16
2 l) x = 5
j) x = 5.5
2. a) x = 0, x = 1.585 k) x = 3

b) x = 0, y = 0 and x = 5, y = 5
2

3. a) x = 19683, x=3 b) x = −1, x = 1
4. a) e−3 , e
5. a) x = 9 b) x = −1, x = 2
6. a) x = −10m b) x = 3
b) x = 3
9m + 3n
b) x = 1, x = 2
7. a) x = 2, 3

8. a) y = 9.638 b) x = 8, x = 1
9. a) x = 1, x = 2 32

b) x = 4

10. a) x  − 1 b) − 7  x  − 4
4 33

c) x  1 d)  − , − 2   (1, )
3
 5
e) (− , − 2)  (4, )
b) x : −3  x  1
11. a) − 5,−2)
b) −1  x 1
12. a) x 1 or x  4

c) x  2 or x  13
4

13. a) (−4, −1)  (3 , ) b) (−4,−1]  (1 ,2]
32 2

14. a) (− ,0)  1, 3  b)  − ,− 4    − 1 ,1  (2, )
2 
 3  2 

15. a)  − 2, 1   (3, ) b) x : x  (− ,−4)   − 2, 3 
2 
 2  

16. a) 1, 7   (6, ) b)  − , 5   (4, )

 2  2

17. a) x : x  2 or x  3.17

18.a) x =  5 , 7  b) x =  −5 , 1  c) x = − 3, − 5, 5, 3
3  24 24 

d) x = 0  e) x = 3 f) x = 1

g) x = 6, − 0.6

19. a) x = −2, 0,1,3 b) x = − 5 ,− 7 ,1, 3 c) x = 25 , 31
 3 6 2   
 7 7 

20. a) ( −, −3)   1 ,   b) (−, −1
 3 

c) − 3 ,   d) ( −, −8)   − 8 ,  
5   3 

e)  −, 4  2,  ) f)  −, − 5    5 ,  
 3   6   2 

g)  3 , 2    2 ,1
 7 3   3

21. x + 4  16 − c ; x = ( − , − 7 )  ( −1,  )

22. a) x : x  −1.45  x  1 b) x  −3 or x  2.37

23. a) (− , −10)   − 10 ,  b) − 1  x  5
 4 
3  2 

24. a) (−,−6]  [0, )

25. a) (−,−8)  (− 8 , ) b) (− ,−7  − 5 ,  
3 3 

26. x = −1, x = 1.465

27. a) x = 4 b) (−, −3)  (−2, −1)  (3,)

28. a) (−, −3) 2,) b) − 3  x  − 4  −8  x  − 3
27 2
29. a) x = 1
b) (−4, −22,3)

30. a) x = 4 b)  −, 1    1 , 7
 2   2 4 

31. x = 9 , y = 5
22

32. a) x =  25 , 31 b) (−, −42,3)
 7 
 7 