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Chapter 8 Inequalities
Chapter 8
8.1
Inequalities
WARM-UP EXERCISE
1. Solve the following equations.
(a) x  4  8
(b) x  3  7
(c) x  5  9
(b) 2x  6
(c)
(b) 5x  8  2  3x
(c) 4  x  3  2x
(b) 2(x  1)  5(x  1)
(c) 4(1  x) 
2. Solve the following equations.
(a) 3x  12
x
 10
5
3. Solve the following equations.
(a) 6x  5  2x  3
4. Solve the following equations.
(a) x  5  3(2x  5)
1
(8x  10)
2
BUILD-UP EXERCISE
[ This part provides two extra sets of questions for each exercise in the textbook, namely Elementary Set and
Advanced Set. You may choose to complete any ONE set according to your need. ]
 Elementary Set
Level 1

1. Determine whether each of the following is correct. If yes, put a ‘  ’ in the box; otherwise
put a ‘  ’.
(a) The symbol ‘’ is an unequal sign.
(b) The symbol ‘’ means less than or equal to.
(c) 8  8
(d) 5  2
1
(e) 0.5 
2
□
□
□
□
□
Ex.8A Elementary Set
Exercise 8A
8.2
New Trend Mathematics S2B — Junior Form Supplementary Exercises
2. Among the following sentences, which of them are descriptions of the inequality x  15?
Which of them are descriptions of the inequality x  15? Write the corresponding letters in
the appropriate circles.
A
x is not less than 15.
B
x is less than or equal to 15.
C
x is at most equal to 15.
D
x is at least equal to 15.
E
x is not greater than 15.
F
x is greater than or equal to 15.
x 15
x 15
3. Match the following sentences with appropriate inequalities.
Twice of y is greater than 5. 
 y  0.5
y is greater than 5. 
 2y  5
Ex.8A Elementary Set
Half of y is greater than 2. 
 2y  5
Twice of y is less than 5. 
 y  5
Twice of y is less than or equal to 5. 

y is less than 0.5. 
y
2
2
 2y  5
4. Represent the solutions to each of the following inequalities graphically.
(a) x  2
3
(b) x  0
2
1
0
1
2
(c) x  1
3
3
3
2
1
0
1
2
3
2
1
0
1
2
3
(d) x  3
2
1
0
1
2
3
3
5. It is given that a  b  0. Fill in the boxes with ‘’ or ‘’ appropriately.
□ b2
(c) 2a □ 2b
(b) a  3
□ (5)  b
(g) 7a □ 7b
(f) a  (1)
(a) a  2
(d)
(e) (5)  a
(h)
a
4
3
a
□ b3
□ b4
□
□ b  (1)
3
b
6. Determine whether each of the following is a solution to the inequality 2x  4.
(a) x  4
(b) x  4
(c) x  2
Chapter 8 Inequalities
8.3
(a) x  1  2
(b) x  4  10
(c) x  5  9
(d) x  3  5
(e) x  1  3
(f) x  4  0
8. In each of the following, form a new inequality by adding the number in the square bracket
to both sides of the inequality such that it is equivalent to the original one.
x
(a) 5x  3
[2]
(b) 4x  2
[9]
(c)
[6]
3
4
Level 2
9. (a) Simplify each of the following inequalities by dividing both sides of the inequality by
the number in the square bracket.
(i) 10x  5
[5]
(ii) 8  16x
[8]
(iii) 8x  4
[4]
Ex.8A Elementary Set
7. Find a solution to each of the following inequalities by trial and error.
(b) Are the inequalities 10x  5, 8  16x and 8x  4 equivalent?
10. In each of the following, if y  5, express the range of values of x by an inequality.
(a) x  2y
(b) x  3y  4
 Advanced Set
Level 1
(c) x  5y

(a) The symbol ‘’ means not less than.
(b) The symbol ‘’ means at least equal to.
(c) 4  4
1
1
(d)   
3
2
(e) 0.02  0.1
2. Express each of the following sentences by an inequality.
(a) 3 times of y is less than 10.
(b) Subtracting 11 from 9 times of y is greater than 3.
(c) Adding 5 to half of x is greater than x.
(d) The sum of x and 5 times of x is less than 9.
(e) Subtracting 3 from half of z is greater than or equal to 3 times of z.
1
(f) The sum of z and of z is at most equal to half of 15.
5
□
□
□
□
□
Ex.8A Advanced Set
1. Determine whether each of the following is correct. If yes, put a ‘  ’ in the box; otherwise
put a ‘  ’.
8.4
New Trend Mathematics S2B — Junior Form Supplementary Exercises
3. Represent the solutions to each of the following inequalities graphically.
(a) x  3
1
(d) x 
2
(b) x  4
(e) x  2.5
(c) x  5
1
(f) x  
4
4. It is given that m  n  0. Fill in the boxes with ‘’ or ‘’ appropriately.
(a) m  (3)
m
2
(d)
□
□ n  (3)
n
2
□ n  (7)
(e) 4m □ 4n
(b) m  (7)
□ 0.5n
□ 6n
(c) 0.5m
(f)
6
m
x
 2 1.
4
(c) x  13
5. Determine whether each of the following is a solution to the inequality
(a) x  13
(b) x  0
Ex.8A Advanced Set
6. Find a solution to each of the following inequalities by trial and error.
1
(a) x  8  10
(b) x  7 
(c) 5x  1  4
2
x
x
(d) 3x  4  5x  6
(e) x   6
(f) 3x   5
2
3
7. In each of the following, form a new inequality by subtracting both sides of the inequality
by the number in the square bracket such that it is equivalent to the original one.
1
(a) 5x  3
[2]
(b) 4x  3  2
[3]
(c) 2x  5x  1
[ ]
5
Level 2
8. Are the following inequalities equivalent to each other? Explain briefly.
10x  6............. (1)
3
x  ................. (2)
5
x 1
  ................ (3)
3 5
9. In each of the following, if y  4, express the range of values of x by an inequality.
y
(a) x 
(b) x  4y  5
(c) x  10  3y
2
8.5
10. Give an example to prove that each of the following sentences is not correct.
1
1
(a) If a  b, then 2  2 .
a
b
x x
(b) For any number x, 
.
3 10
1
1
(c) If a  b  0 and b  c  0, then
.

ab ac
Exercise 8B
 Elementary Set
Level 1
Ex.8A Advanced Set
Chapter 8 Inequalities

1. Solve the following inequalities and represent the solutions graphically.
(a) x  1  3
(b) x  2  5
(c) x  4  5
(d) x  5  4
(e) x  6  3
(f) x  3  8
2. Solve the following inequalities and represent the solutions graphically.
(b) 4  x  5
(d) 5  x  1
(e) 6  5  x
1
2
1
1
(f)
x
3
3
(c) 4  x  
3. Solve the following inequalities and represent the solutions graphically.
(a) 3x  9
x
(d)
1
2
(b) 5x  10
x
(e) 3  
4
(c) x  1
3x
(f) 5  
2
4. Solve the following inequalities and represent the solutions graphically.
(a) 2x  1  5
(b) 3x  2  7
(d) 1  5x  6
(e) 7x  5  9
(c) 1  3x  10
1 1
(f) 4 x  
2 2
5. Solve the following inequalities and represent the solutions graphically.
(a) 2x  1  x  3
(b) 2x  1  x  4
(c) 4  x  2x  3
(d) x  2  4  3x
(e) 10  2x  x  1
(f) 5  3x  1  3x
Ex.8B Elementary Set
(a) 2  x  3
8.6
New Trend Mathematics S2B — Junior Form Supplementary Exercises
Level 2
Ex.8B Elementary Set
6. Solve the following inequalities and represent the solutions graphically.
1
1
(a) 5(x  8)  10
(b) (4 x  1)  1
(c) 3  (2 x  3)
3
5
7. Solve the following inequalities and represent the solutions graphically.
1
1
(a) 4(x  3)  5(x  2)
(b) 3(x  7)  4(3x  1)
(c) (2 x  5)  (5x  3)
8
4
8. Solve the following inequalities and represent the solutions graphically.
x
2x
x
3x
3x 4 x
(a)
(b)  1 
(c) 6   1 

 3
5 15
3
9
3
4
 Advanced Set
Level 1

1. Solve the following inequalities and represent the solutions graphically.
(a) x  5  8
1
1
(d)  x 
2
2
(b) x  1  1
(e) 8  x  7
(c) x  10  4
1
1
(f)
x
4
4
Ex.8B Advanced Set
2. Solve the following inequalities and represent the solutions graphically.
x
(a) 4x  8
(b) x  9
(c)
 10
5
x 1
x
(d) 8  4x
(e) 9  
(f)

5
7 14
3. Solve the following inequalities and represent the solutions graphically.
4
1
(a) 5x  8  3
(b) 2  5x  7
(c)
 2x  
5
5
3x
x
(d) x  7  3x  5
(e) 6  7x  3  4x
(f) 5 
  3
2
2
Level 2
4. Solve the following inequalities and represent the solutions graphically.
1
(a) 3(x  2)  2
(b) 5( x  1) 
(c) 3(2x  1)  1
10
x
1
2
3 1
(d) 3(  4)  9
(e) (9 x  1)  
(f) 3  (2 x  3)
3
3
3
5 4
8.7
Chapter 8 Inequalities
5. Solve the following inequalities and represent the solutions graphically.
x
1
(a) x  3  4(x  2)
(b) 2x  3  5(2x  1)
(c)
 3  (27  2 x)
2
3
Ex.8B Advanced Set
6. Solve the following inequalities and represent the solutions graphically.
1
1
(a) 6(3  x)  4(2x  3)
(b)
(32 x  16)   (8 x  7)
16
4
2x 1
x
(c)
 (12  3x)    5
3 4
3
7. Solve the following inequalities and represent the solutions graphically.
1
1
5( x  3)
1
3
(a) 1 x  x  4
(b) 2 x  1  x
(c)
 2  4x  1
3
6
7
2
4
8. (a) Solve the inequality
7
9
m  12  m .
4
4
(b) Using the result of (a), solve the inequality
7
9
(3x  2)  12  (3x  2) .
4
4
Exercise 8C
 Elementary Set
Level 1

1. Fill in the blanks.
(b) If y is less than 10, i.e.
, then the maximum integral value of y is
2. Fill in the blanks.
(a) Given that twice of x is greater than 4, it can be expressed by an inequality
2x  4

 The minimum integral value of x is
.
(b) Given that 3 times of y is less than 9, it can be expressed by an inequality


 The maximum integral value of y is
.
.
.
Ex.8C Elementary Set
(a) If x is greater than 5, i.e. x  5, then the minimum integral value of x is
8.8
New Trend Mathematics S2B — Junior Form Supplementary Exercises
3. Fill in the blanks.
Given that the sum of two consecutive numbers x and x  1 is greater than 19, it can be
expressed by an inequality
x  (x  1) 

x
 The minimum integral value of the smaller number is
.
Ex.8C Elementary Set
4. If the airtime of mobile phone consumed by Wyman this month was less than 525 minutes,
find the maximum airtime consumed by him this month.
5. If half of z is less than or equal to 5, find the maximum integral value of z.
6. The sum of two consecutive odd numbers is greater than 24. Find the minimum value of the
larger number.
Level 2
7. Cherry spent $x yesterday and $3x today on shopping. If she spent not more than $100 on
shopping in these two days, how much did she spend at most today?
8. The daily pocket money of Sam from Monday to Friday this week was $x, and he got $30
more on both Saturday and Sunday. If the pocket money of Sam this week was not less than
$200, at least how much daily pocket money did he get from Monday to Friday?
 Advanced Set
Level 1

1. If half of a is less than 10, find the maximum integral value of a.
Ex.8C Advanced Set
2. If the sum of 4 and 5 times of x is greater than or equal to 10, find the minimum integral
value of x.
3. The sum of two consecutive even numbers is greater than 17. Find the minimum value of
the larger number.
Level 2
4. For two consecutive multiples of 4, the larger number is less than twice of the smaller one
by at least 16. Find the minimum values of the two numbers.
Chapter 8 Inequalities
8.9
6. For the first term examination, Wendy scored 65 in Chinese Language and x in English
Language. Her score in Mathematics is twice of that in English Language. If the average
score of Wendy in these three subjects was not less than 70, what was her minimum score
in English Language?
7. Mr. Chan is going to buy 3 Chinese books and several English books. The prices of these
Chinese and English books are $40 and $55 each respectively. If Mr. Chan only has $550,
at most how many English books can he buy?
8. On a highway, car A and car B travel in an opposite direction from each other from the
same starting point. The speeds of cars A and B are 75 km/h and 85 km/h respectively. How
long (in minute) will it take for cars A and B to be at least 30 km apart?
CHAPTER TEST
(Time allowed: 1 hour)
Section A (1) [ 3 marks each ]
1. Is x  3 a solution of the inequality x  4  5?
2. Express the sentence ‘3 times of x is greater than 7’ by an inequality.
3. Represent the solutions of the inequality x  7 graphically.
4. Write down the inequality represented by the following figure.
4
3
2
1
0
1
5. Solve the inequality x  2  7, and represent the solutions graphically.
6. Solve the inequality 3x  12, and represent the solutions graphically.
Section A (2) [ 6 marks each ]
7. Solve the inequality 3x  2  7, and represent the solutions graphically.
Ex.8C Advanced Set
5. The lengths of the sides of a triangle are x cm, 12 cm and 2x cm, where x is an integer. If
the perimeter of the triangle is not less than 40 cm, find the minimum length of the shortest
side.
8.10
New Trend Mathematics S2B — Junior Form Supplementary Exercises
8. Solve the inequality 2(x  5)  8, and represent the solutions graphically.
9. Solve the inequality x  3  2x  1, and represent the solutions graphically.
10. Solve the inequality
1
(10 x  4)  x  3 , and represent the solutions graphically.
5
Section B
11. (a) Solve the inequality 30y  20(8  y)  195, and represent the solutions graphically.
(5 marks)
(b) Karen is going to buy altogether 8 gift sets A and B for her friends. The prices of gift
sets A are $30 each and those of gift sets B are $20 each. It is known that Karen wants
to buy as many gift sets A as possible for not more than $195.
(i) According to the result of (a), how many gift sets of each kind should Karen buy?
(ii) How much does Karen spend on the gift sets?
(8 marks)
Multiple Choice Questions [ 3 marks each ]
12. Which of the following figures represents
the inequality x  2?
A.
0
1
2
14. Which of the following figures
represents the inequality with a solution
of x  1?
A.
3
B.
0
1
2
1
2
0
1
2
3
□
13. The following figure represents the
inequality
2
1
0
1
1
0
1
2
1
0
1
2
1
0
1
2
□
15. Which of the following figures
represents all numbers greater than 2?
A.
2
B. x  1.
D. x  1.
2
D.
A. x  1.
C. x  1.
1
C.
3
D.
0
B.
3
C.
0
1
3
2
1
0
3
2
1
0
B.
□
Chapter 8 Inequalities
20. If a  b, which of the following must be
correct?
C.
3
2
1
0
3
2
1
0
A. a  3  b  3
D.
□
B. a  3  b  3
C. a  3  b  3
□
D. a  3  b  3
16. Which of the following cannot be a
solution of the inequality represented by
the following figure?
5
4
3
2
1
1
B. 2a  2b
C. 2a  2b
B. x  0
D. x  3
□
A. x  5
□
A. x  3  8
B. x  3  2
D. x  3  8
A. 1
B. 1
□
D. b 2
18. Which of the following is equivalent to
the inequality x  5?
C. 3x  15
a b
 , which of the
c c
following is a possible value of c?
22. Given that a  b and
C. a 2
B. x  5
D. x  5
23. Solve the inequality 8  2x  5.
A. x  3
B. x  3
3
C. x 
2
D. x  
□
19. Which of the following is equivalent to
the inequality 8x  24?
A. x  3
B. x  3
D. 3x  1
□
□
3
2
24. Solve the inequality 
C. x  3
□
D. 2a  2b
17. Which of the following inequalities
shows the meaning of ‘the least value of
x is 5’?
C. x  5
21. If a  b, which of the following must be
correct?
A. 2a  2b
0
A. x  2
C. x  1
8.11
A. x  12
4
B. x  
3
4
C. x  
3
D. x  12
x
 4.
3
□
8.12
New Trend Mathematics S2B — Junior Form Supplementary Exercises
25. There are 25 boys in S1A. If the number
of boys is at least 6 more than that of
girls, at most how many girls are there in
S1A?
A. 18
B. 19
C. 20
D. 31
□
26. For two consecutive numbers, the sum of
the smaller number and twice of the
larger number is not less than 48. Find
the minimum value of the larger number.
A. 15
B. 16
C. 17
D. 18
□