TUTORIAL CHAPTER 2 : EQUATION, INEQUALITIES & ABSOLUTE VALUES

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

CHAPTER 2: EQUATIONS, INEQUALITIES AND ABSOLUTE VALUES

TUTORIAL 1 OF 5

1. Solve

(a) −2 = 1 (d) 2(22x ) − 5(2x ) + 2 = 0
(e) 32x+1 − 26(3x ) − 9 = 0
4x 3 9 (f) x2 10x − x 10x = 2 10x
(g) ex −12e−x −1 = 0
(b) 7x2 − 496−2x = 0
(c) x +1 + 12 − x = 13 + 4x
(c) ( 4x )x = 4 8x

2. Solve

(a) x + 5 = 7 − x
(b) 3 1− x = 1+ 3x − 2

3. Solve (g) log3 x − 2logx 3 = 1
(a) (5x )(5x−1) = 10
(h) 2ln x = ln 3 + ln(6 − x)
(b) ex ln 3 = 27
(c) e2 ln x = 9 (i) 9logx 5 = log5 x
(d) 3e−2x = 75 (j) eln(1−x) = 2x
(k) log2(2x+1 − 32 2) = x
(e) 2log( x − 2) = log(2x − 5) (l) x6e−4ln x = 2x +15
(f) log16 x = (log4 x)2, x  1

ANSWER TUTORIAL 1

1. (a) 216 (b) -6,2 (c) − 1 ,2 (d) -1,1
(e) 2 (f) -1,2 2
(d) -1.6094
2. (a) -2,-9 (g) ln 4 (h) 3
(l) 5
3. (a) 1.2153 3 (c) 3
(e) 3
(b)
1
4
(i) 125,
(b) 3 (c) 3
125 (f) 2 (g) 9, 1

1 11 3
11
(j) (k)
2
32

Page 1 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

TUTORIAL 2 OF 5

1. Solve the following inequalities. Write down the answer in interval form.

(a) 3x −1  2( x − 5) (b) − 7  3 − 4x  6

(c) x − 4  3 − x (d) 7 − 3x  13
(e) 1− 7x  x + 3 (f) x + 3  3x − 5

2. Find the ranges of values of x that satisfy the following inequalities using real number

line. Write down the answer in solution set form.

(a) ( x +1)( x − 2)  0 (b) (x + 1)(x − 4)  0

(c) 2x2 − 3x − 2  0 (d) x 2 − 2x − 15  0

3. Find the ranges of values of x that satisfy the following inequalities using table of signs.

Write down the answer in interval form.

(a) 5x2  3x + 2 (b) ( x −1)2  9

(c) x2 − 4x  −3 (d) ( x −1)( x − 2)  0

(e) 4x2  1 (f) (1− x)(4 − x)  x +11

4. Find the range of values of x for which 3x + 4  x2 − 6  9 − 2x .

ANSWER TUTORIAL 2

1. a) (− , − 9) b)  − 3 , 5 c)  − , 7 
4 2   2

d) (− 2, ) e)  − , − 1  f) (4, )
 4

2. a) x : x  −1  x  2 b)  x : −1  x  4 

c)  x : − 1  x  2 d) x : x  −3  x  5
 2 

3. a)  − , − 2   (1, ) d) 1, 2

 5 e)  − 1 , 1 
 2 2
b) (− , − 2)  (4, )
c) (− , −1  3, ) f) (− , −1)  (7, )

4. − 5  x  −2 or (− 5, −2)

Page 2 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

TUTORIAL 3 OF 5

1. Find the range of values of x for each of the following inequalities. Write down the

answer in interval form.

(a) x + 5  0 (b) 8 − 4x  1− x
2x −1 x+5

(c) x − 2  3 (d) x − 3  1
x−5 2x

2. Find the possible values of x for which 11.
x −1 x +1

3. What values of x satisfy the inequality 2x  x ?
x +1

4. Solve the given equations by using definition. Write down the answer in solution set
form.

(a) x +1 = 2 (b) 12x +1 = 3 (c) 2x − 5 = 3
x−3 24

(d) x2 − 7 = 2 (e) 4 = 8
x+2

5. Solve the following equations by squaring both sides. Write down the answer in
solution set form.

(a) 4 − 3x = 5x + 4 (b) x − 2 = 10 − 3x (c) x + 2x = 3

ANSWER TUTORIAL 3

1. − 5, 1  c)  5, 13 
a) 2  2
b)
( − 5,  ) d)  −1, 0  (2, )

2 

2. x : −1  x  1
3. x : x  −1 0  x  1

4. a) x =  5 , 7   d) x = − 3, − 5, 5, 3
 3 
 

b) x =  −5 , 1 e) x =  − 5, −3 
 24   2 2 
 24   

c) x = 1, 4 

5 a) x =  0, − 4  b) x = 3, 4  c) x = 1, 3

Page 3 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

TUTORIAL 4 OF 5

1. Solve the following absolute value inequalities by using the basic definition. Write

down the answer in interval form.

(a). 2x − 3  1 (b) x  5 (c) x −1  2
36 x+3

(d) x  2 (e) 2x  1 (f) x2 − 6x + 4  4
x+4 x+3

2. Find the range of values of x that satisfy the following inequalities by using basic

definition. Write down the answer in solution set form.

(a) 3x − 4  2x + 1 (b) 2 x  3x −10 (c) 2x + 1  3x + 2

3. Find the possible value of x that satisfy the following inequalities by squaring both

side. Write down the answer in interval form.

(a) x − 3  2x + 5 (b) 2x − 5  10 − 3x

(c) x − 4  2x − 6 (d) x + 4  3x + 2

4. Find the value of k if −2  x  3 is equivalent to x − k  5 .
2

5. Solve the following inequalities

(a) 9x + 4  12 (b) 1 + log2 x − 2 log x 2  0
x

6. 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.

ANSWER TUTORIAL 4

1. a) ( − , 1    2,  ) b)  − 5 , 5 c)  − 7, −3)   − 3, − 5 
 2 2   3 

d) ( − , − 8 )   − 8 ,  e)  −1, 3  f)  0, 2    4, 6 

3 

2. a) x = x  3 or x  5 b) x =  x  10  c) x = x  − 3 
 5   5 

3. a) x =  − 8, − 2  c) x = (− ,2  10 ,  
 3  3 

b) x = ( − , 3 )  ( 5,  ) d) x =  − 3 , 1 
 2 

4. k = 1
2

5. a) x : x  0 b)  1 ,1  (2, )

4 

6. x + 4  16 − c

x =  x  −7  x  −1 or x = ( − , − 7 )  ( −1,  )

Page 4 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

TUTORIAL 5 of 5

PAST SEMESTER EXAMINATION (PSE)

SESSION 2013/2014

1. Find the value of x which satisfies the equation log9 x = (log3 x)2, x  1 [7]
[7]
Answer: x = 3

2. Solve the equation 2x−2 − 2x+1 = 2x − 23

Answer: x = 3 or x = 2

3. (a) Find the solution set of 2 − 3x  x + 3 [8]

(b) If x +1  0, show that (ii) 2x −1  2 [3]
(i) 2x −1  0 x +1 [4]

Answer: a) x : − 1  x  5
4 2 

SESSION 2014/2015 [6]
4. Solve the equation 3x + 33−x = 12. Answer: x = 2 or x = 1
5. Solve the inequality 1  1 .
6− x x −1 [6]

6. (a) Solve the following equation 6x2 + x −11 = 4. Answer: x :1  x 7x  6
 2 

[6]

(b) Find the solution set for the inequality 2 −  x + 2   5. [7]
 x − 4 

Answer: a) x = − 5 , − 7 ,1, 3 b)  x : x  5  x  4
362  2 

SESSION 2015/2016

7. Solve the equation 2 + log2 x = 15logx 2. [7]
[6]
Answer: x = 8, x = 1 [9]
32

8. (a) Solve the inequality x −1  2
x+3

(b) Show that 2x  42x = 22x . Hence, find the interval for x so that
8x

( )2x  42x−132x + 36  0 .
8x

Answer: a) ( −7, −3)   −3, − 5 b) Shown. (−, 2 3.17, )
 3 

Page 5 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

SESSION 2016/2017 [6]
[6]
Determine the value of x which satisfy the equation 32x=1 = 4(3x ) − 9 [7]

9. Answer: x = 1 or x = 2 [5]
[8]
10. (a) If 7 − 3 5 = x − y , determine the values of x and y

(b) Solve the equation log2 x − log4 (3x + 4) = 0

Answer: x = 9 , y = 5 or x = 5 , y = 9 b) x = 4
22 22

11. (a) Solve the following equation

3 = 7, x  4.
x−4

(b) Find the solution set for the inequality b) x : −  x  −4  2  x  3
−4 − x  x + 4, x  3.
x−3
Answer: a) x = 31 or x = 25
77

SESSION 2017/2018 [6]
[9]
12. Solve the equation 32x+1 − (16) 3x + 5 = 0
[4]
Answer: x = −1 or x = 1.465 [9]

13. Solve the equation 3log9 x = (log3 x)2 .

Answer: x = 1 or x = 5.196

14. Find the interval of x for which the following inequalities are true.

(a) 5 −1  0 (b) 3x − 2  2
x+3 2x +3

Answer: a) x : x  −3  x  2 b)  −8, − 3    − 3 , − 4 
 2   2 7 

SESSION 2018/2019

15. (a) Solve 6x +1 − x = 3 [6]
[7]
(b) Determine the solution set of x which satisfies the inequality
[6]
2 x [6]
x +1 x +3

Answer: a) x = 4 b) x : x  −3  −2  x  −1 x  3

16. Solve

 27 2  25 4 x  9  x−3  625 2
 125   9   25   81 
(a)  = 

(b) 1  8
4− 2x x

Answer: a) x=2 b)  x : x  0  32  x  2
 17

Page 6 of 7

PERLIS MATRICULATION COLLEGE
TUTORIAL & PSE SM015 Session 2021/2022

SESSION 2019/2020

17. Solve the following:

( )(a) 3 52x 1 x+1 [5]

+ 252 = 200 [5]

(b) x + 4  x2 + x  12 Answer: a) x = 1 b) (−4, −22,3)

18. Solve the following: [6]

(a) log2 2x = 2 log4 ( x + 4) [7]

(b) 2 x − 3  1
2x −1

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

SESSION 2020/2021 [7]
19. Determine the solution set of the following inequalities.
(a) 4x  6 b) x : x  0  x  3
4−x

(b) x − 6  3 x − 2

Answer: a)  12 , 4 
 5 

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