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1. Determine whether each of the following statements 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) The symbol ‘’ means at least equal to.
(d) 8  8
(e) 5  2
(f) 0.5 
1
2
(g) 0.1  0.02
□
1
1

2
3
□
(h) 
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
8.15
 2009 Chung Tai Educational Press. All rights reserved.
3. Match the following sentences with appropriate inequalities.
Twice of y is not less than 5. 
 y  5
y is greater than 5. 
 2 y  5
Half of y is greater than or equal to 5. 
 2y  5
Twice of y is not greater than 5. 
 y  5
Twice of y is less than or equal to 5. 

y is less than 5. 
y
 5
2
 2y  5
4. Represent the solutions of each of the following inequalities graphically.
(a) x  11
7
(b) x  9
8
9
10
11
12
13
(c) x  10
13
4
5
6
7
8
9
10
14
13
12
11
10
9
(d) x  13
12
11
10
9
8
7
15
5. Write down the inequality in x shown by each of the following graphical representations.
(a)
(b)
13
(c)
1
14
15
16
17
18
19
0.9 0.8 0.7 0.6 0.5 0.4
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 2009 Chung Tai Educational Press. All rights reserved.
1
1
2
0
1
2
1
3
2
2
2.2
2.4
2.6
2.8
3
3.2
3.4
(d)
6. Express each of the following sentences in an inequality.
(a) The product of 3 and y is less than 10.
(b) The product of 9 and y minus 11 is greater than 3.
(c) Adding 5 to half of x is greater than x.
(d) The sum of x and 5 times x is less than 9.
(e) Half of z minus 3 is greater than or equal to the product of 3 and z.
1
(f) The sum of z and
of z is at most equal to half of 15.
5
7. Determine whether each of the following is a solution of the inequality 2 x  1  4 .
(a) x  4
(b) x  1
(c) x  4
8. Determine whether each of the following is a solution of the inequality
(a) x  13
x
 2  1.
4
(b) x  0
(c) x  13
9. Given that a  b  0 , fill in the blanks appropriately with ‘’ or ‘’.
(a) a  2
(c) 2b
□ b2
□ 2a
(e) (5)  a
(g) 7a
□ (5)  b
□ 7b
(i) 1  6b
□ 1  6a
(b) a  3
(d)
a
4
□ b3
□
b
4
(f) b  (1)
□
a  (1)
(h) 4a  11
□
4b  11
a
9
5
□
b
 9
5
(j) 
8.17
 2009 Chung Tai Educational Press. All rights reserved.
10. Add the same number to or subtract the same number from both sides o f each inequality below, and
obtain an inequality in the form of x  a or x  b .
(a) x  8  10
(b) x  13  4
(c) 21  6  x
(d) 7  2  x
11. Multiply both sides of each inequality below by the same number, and obtain an inequality in the form
of x  a or x  b .
x
(a) 196  7 x
(b)   13
11
5x
(c) 8 x  60
(d) 75 
4
12. If x  10 , express the ranges of values of a, b and c in inequalities.
(a) a  4 x
4 x  17
(c) c  
19
(b) b  4 x  17
13. If x  6 , express the ranges of values of a, b and c in inequalities.
(a) a  x  8
28  3( x  8)
(c) c 
7
(b) b  3( x  8)
x x
 ’ is correct. To point out that the opinion of Patricia is
2 5
x x
wrong, give a value of x such that
 .
2 5
14. Patricia thinks that ‘for any number x,
1
1
’ is correct. To point out that the opinion of Dickson is

ab ac
1
1
wrong, give a set of values of a, b and c such that a  b  c and
.

ab ac
15. Dickson thinks that ‘if a  b  c , then
8.18
 2009 Chung Tai Educational Press. All rights reserved.
16. Solve each of 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
17. Solve each of the following inequalities and represent the solutions graphically.
(a) 2  x  3
(c) 4  x  
(b) 4  x  7
1
2
(e) 6  5  x
(d) 5  x  1
(f)
1
1
x
3
3
18. Solve each of the following inequalities and represent the solutions graphically.
(a) 3x  9
(c) x  1
(e) 3  
x
4
(b) 5 x  10
x
(d)
1
2
3x
(f) 5  
2
19. Solve each of the following inequalities and represent the solutions graphically.
(a) 2 x  1  5
(b) 3 x  2  7
(c) 1  3 x  10
(d) 1  5 x  6
1 1
(f)  4 x  
2 2
(e) 7 x  5  9
20. Solve each of the following inequalities and represent the solutions graphically.
(a) 2 x  1  x  3
(b) 2 x  1  x  4
(c) 4  x  2 x  3
(d) x  2  4  3 x
(e) 10  2 x  x  1
(f) 5  3 x  1  x
8.19
 2009 Chung Tai Educational Press. All rights reserved.
21. Solve each of the following inequalities and represent the solutions graphically.
4
1
(a) 5 x  8  3
(b) 2  5 x  7
(c)
 2x  
5
5
3x
x
(d) x  7  3x  5
(e) 6  7 x  3  4 x
(f) 5 
  3
2
2
22. Solve each of 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)
5
3
23. Solve each of the following inequalities and represent the solutions graphically.
1
2
1
(a) 3( x  2)  2
(b) 5( x  1) 
(c) (9 x  1)  
3
3
10
24. Solve each of 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)  (5 x  3)
8
4
25. Solve each of the following inequalities and represent the solutions graphically.
7 x  32
66  28 x
3x  80
(a) x 
(b)
(c) 4 x 
 2x
15
9
4
26. Solve each of the following inequalities and represent the solutions graphically.
x
2x
x
3x
3x 4 x
(a)
(b)  1 
(c) 6   1 

 3
3
9
3
4
5 15
27. Solve each of the following inequalities and represent the solutions graphically.
1
1
2x 1
x
5( x  3)
(a)
(b)
(c)
(32 x  16)   (8x  7)
 (12  3x)    5
 2  4x  1
16
4
3 4
3
7
28. (a) Solve the inequality
7
9
m  12  m .
4
4
(b) Using the result of (a), solve the inequality
8.20
 2009 Chung Tai Educational Press. All rights reserved.
7
9
(3x  2)  12  (3x  2) .
4
4
29. Find the smallest value and smallest integral value of x of each of the following inequalities.
x
x4 x7
(a) 2( x  5)  5( x  3)
(b)  5  3x
(c)

2
5
3
30. Find the largest value and largest integral value of x of each of the following inequalities.
2 x  1 3x  5
(a) 3x  7( x  1)
(b) 3( x  2)  2( x  5)
(c)

3
2
31. 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 smallest value of the smaller number is
.
32. The sum of two consecutive odd numbers is greater than 24. Find the smallest value of the larger
number.
33. The sum of two consecutive even numbers is less than 57. Find the greatest value of the larger
number.
34. For two consecutive multiples of 4, the larger number is less than twice of the smaller number by at
least 16. Find the smallest values of the two numbers.
35. If the difference between two integers is 11, and their sum is less than 135, find the greatest value of
the smaller integer.
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 2009 Chung Tai Educational Press. All rights reserved.
36. Cherry went out for shopping yesterday and today, and the amount she spent today is three times the
amount she spent yesterday. If she spent not more than $300 on shopping i n these two days, at most
how much did she spend today?
37. 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.
38. Sam received the same amount of the daily pocket money from Monday to Friday this week, and he
got $30 more on both Saturday and Sunday. If the pocket money of Sam this week was not less than
$550, at least how much daily pocket money did he get from Monday to Friday?
39. For the first term examination, Wendy scored 63 in Chinese 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?
[ Hint: Average score 
Total score
]
Number of subjects
40. 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 mos t how many
English books can he buy?
41. 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 min) will it take
for cars A and B to be at least 40 km apart?
42. A store bought 250 eggs for $200 and found that 25 of them were cracked. The rest of the eggs will be
sold at a percentage profit of not greater than 35%. Find the maximum sell ing price of each egg.
43. The ingredients of the nuts of Brand X, Brand Y and Brand Z in percentage are as follows:
Brand
X
Y
Z
Walnut
30%
25%
55%
Almond
30%
50%
30%
Hazelnut
40%
25%
15%
Ingredients
How many kg of nuts from Brand Z should be added to 0.5 kg of nuts from Brand X and 1 kg of nuts
from Brand Y, such that there are not less than 20% of hazelnuts in the mixture?
8.22
 2009 Chung Tai Educational Press. All rights reserved.