F.3 Mathematics
M.C. Revision Exercise – Ch.1 Linear Inequalities in One Unknown
<<Solutions>>
Multiple-choice Questions
13043001
Which of the following represents the statement ‘The result of adding half of y to 12 is not greater
than 20’?
y  12
2
y  12
B.
2
y
 12
C.
2
y
 12
D.
2
-- ans --
A.
≤
20
≥
20
≤
20
≥
20
Solution:
The answer is C.
-- ans end --
13043002
Which of the following represents the inequality ‘u + v ≥ 3(s + 1)’?
A. The result of adding u to v is greater than the result of adding 1 to three times s.
B. The result of adding u to v is not less than three times the result of adding s to 1.
C. The result of adding u to v is not less than the result of adding 1 to three times s.
D. The result of adding u to v is greater than three times the result of adding s to 1.
-- ans -Solution:
The answer is B.
-- ans end --
13043003
Which of the following represents the solutions of x ≥ 4?
A.
C.
B.
D.
P. 1
-- ans -Solution:
The answer is B.
-- ans end --
13043004
Using x as the variable, which of the following satisfies the inequality as shown in the above figure?
A.
B.
x = 2
x = 1.2
C.
x =
15
13
D. x = 0
-- ans -Solution:
The answer is D.
-- ans end --
13043005
Which of the following inequalities with solutions include 3 but exclude –2?
A. x > 3
B. x > 1
C. x < 4
D. x < 3
-- ans -Solution:
The answer is B.
-- ans end -13043006
If x ≥ y, which of the following is incorrect?
A. 3x + 5 ≥ 3y + 5
P. 2
x  2 ≥ y 2
2  3x ≥ 2  3y
x
y
2 ≥ 2
D.
3
3
-- ans -B.
C.
Solution:
The answer is C.
x ≥ y
3x ≥ 3y
3x ≤ 3y
2  3x ≤ 2  3y
 C is incorrect.
-- ans end --
13043007
If m ≥ n > 0, then
2
2
A.
≥
.
m
n
2
2
B.
> .
m
n
2
2
C.
≤
.
m
n
2
2
D.
< .
m
n
-- ans -Solution:
The answer is C.
m ≥ n
1
1
≤
m
n
2
2
≤

m
n
-- ans end --
13043008
If 0 < x < y and z is a negative number, then
A. xz < yz.
B. xz2 < yz2.
C.
x(z  4) < y(z  4).
P. 3
z
z
 .
x
y
-- ans --
D.
Solution:
The answer is B.

z < 0

z2 > 0
x < y
 xz2 < yz2
-- ans end --
13043009
If 2x ≤ 3, which of the following is incorrect?
A. 4x ≤ 2x + 3
1  2x ≤ 2
2x + 3 ≤ 6
2x
D.
≤ 1
3
-- ans -B.
C.
Solution:
The answer is B.
2x ≤ 3
2x ≥ 3
1  2x ≥ 3 + 1
1  2x ≥ 2
 B is incorrect.
-- ans end --
13043010
If 4a < 3 and b > 3, then which of the following must be correct?
I. 4a < b
b
II. a 
4
III. a < b
A. II only
P. 4
B.
C.
I and II only
I and III only
D. I, II and III
-- ans -Solution:
The answer is D.
I:  4a < 3 and 3 < b
 4a < b
 I must be correct.
II: According to I,
4a < b
b

a 
4
 II must be correct.
III:  b > 3 > 0
b
 b 
4
According to II,
a < b
 III must be correct.
 I, II and III must be correct.
-- ans end --
13043011
If p > q, which of the following must be correct?
I.
II.
III.
A.
p < q
p > pq
p2 > q2
I only
B. II only
C. I and II only
D. I and III only
-- ans -Solution:
The answer is A.
I:
p > q


p < q
I must be correct.
P. 5
II:
If p = 1 and q = 0,
1 > 0
But p  q = 1  0
=1
 II may not be correct.
III: If p = 2 and q = 3,
2 > –3
But p2 = 4 and q2 = 9
i.e.
p2 < q2
 III may not be correct.
 Only I must be correct.
-- ans end --
13043012
Given that y = 3x + 5. If x ≥ 4, which of the following inequalities represents the range of values
of y?
A. y ≥ 7
B. y ≥ 12
C. y ≥ 17
D. y ≥ 23
-- ans -Solution:
The answer is C.
x ≥ 4
3x ≥ 3(4)
3x + 5 ≥ 12 + 5
3x + 5 ≥ 17
 y = 3x + 5
 y ≥ 17
-- ans end --
13043013
Given that x ≥
1
, which of the following inequalities represents the range of values
3
of 3x + 1?
A. 3x + 1 ≥ 0
B. 3x + 1 ≥ 2
P. 6
C.
D.
3x + 1 ≥ 2
3x + 1 ≤ 0
-- ans -Solution:
The answer is B.
1
x ≥
3
3x ≥ 1
 3x + 1 ≥ 2
-- ans end --
13043014
1
Given that y  ( x  4) . If y ≤ 2, find the range of values of x.
3
A. x ≥ 10
B. x ≥ 2
C. x ≤ 2
D. x ≤ 10
-- ans --
Solution:
The answer is D.
1
y  ( x  4)
3
3y = x  4
x = 3y + 4
 y ≤ 2
3y ≤ 3(2)
3y + 4 ≤ 6 + 4
3y + 4 ≤ 10
 x = 3y + 4
 x ≤ 10
-- ans end --
13043015
Which of the following inequalities with solutions include 4?
I. 3x  2 ≥ 2x + 1
II. x + 3 ≥ 3x  11
III. 2x + 1 > x + 5
P. 7
A.
B.
I only
II only
C. I and II only
D. I and III only
-- ans -Solution:
The answer is C.
I:
3x  2 ≥ 2x + 1
3x  2x ≥ 1 + 2
x ≥ 3
 4 satisfies the inequality in I.
x + 3 ≥ 3x  11
3 + 11 ≥ 3x  x
14 ≥ 2x
7 ≥ x
 4 satisfies the inequality in II.
III: 2x + 1 > x + 5
II:

2x  x > 5  1
x > 4
 4 does not satisfy the inequality in III.
Only the solutions of the inequalities in I and II include 4.
-- ans end --
13043016
The three Mathematics test results of Macy are 72, 85 and x respectively. If the average mark of the
three tests is not greater than 80, then which of the following inequalities can be used to find the
range of values of x?
A.
72  85  x
≥ 80
3
B.
72  85  x
> 80
3
72  85  x
≤ 80
3
72  85  x
D.
< 80
3
-- ans --
C.
Solution:
The answer is C.
P. 8
-- ans end --
13043017
Solve the inequality 2(x  3) ≥ 5x + 6.
A. x ≥ 4
B.
C.
x ≥ 4
x ≤ 4
D. x ≤ 4
-- ans -Solution:
The answer is D.
2(x  3) ≥ 5x + 6
2x  6 ≥ 5x + 6
6  6 ≥
12 ≥
4 ≥
i.e.
x ≤
-- ans end --
5x  2x
3x
x
4
13043018
Solve the inequality
A.
B.
C.
D.
x
x
x
x
<
>
<
>
2x  3
 3.
5
6
6
9
9
-- ans -Solution:
The answer is A.
2x  3
 3
5
2x + 3 < 15
2x < 15  3
2x < 12
 x < 6
P. 9
-- ans end --
13043019
Solve the inequality
A. y ≤
B. y ≥
C. y ≤
D. y ≥
-- ans --
y 8
2
(2 y  5) ≤
7.
3
6
5
5
10
10
Solution:
The answer is C.
y 8
2
(2 y  5) ≤
7
3
6
4(2y  5)
8y  20
8y  y
7y

y
-- ans end --
≤
≤
≤
≤
≤
y + 8 + 7(6)
y + 8 + 42
50 + 20
70
10
13043020
Which of the following is NOT a solution of the inequality
I.
II.
III.
A.
B.
C.
x = 2
x = 4
x=0
II only
III only
I and II only
D. I and III only
-- ans -Solution:
The answer is A.
5x
 3 ≤ 2x + 1
6
5x – 18 ≤ 12x + 6
–18 – 6 ≤ 12x – 5x
–24 ≤ 7x
P. 10
5x
 3 ≤ 2x + 1?
6
3
i.e.

3
≤ x
7
x ≥ 3
3
7
4 is not a solution of the inequality
5x
 3 ≤ 2x + 1.
6
-- ans end --
13043021
Which of the following inequalities has solution x ≥ 4?
x6
A. x ≥
3
B. 3(5  x) + 1 ≤ 4
C. 4  3x ≥ 16
2
( x  7) ≥ 3
D.
5
-- ans -Solution:
The answer is B.
x6
A: x ≥
3
3x ≥ x + 6
2x ≥ 6
x ≥ 3
B:
C:
3(5  x) + 1
15  3x
12
4
i.e.
x
≤
≤
≤
≤
≥
4
3
3x
x
4
4  3x ≥ 16
3x ≥ 12
x ≤ 4
D:
2
( x  7) ≥ 3
5
2x + 14 ≥ 15
2x ≥ 1
1
x ≥
2
 The answer is B.
-- ans end -P. 11
13043022
Find the smallest integer satisfying the inequality
2x  3
≥ 2.
4
A. 3
B. 2
C. 2
D. 3
-- ans -Solution:
The answer is B.
2x  3
≥ 2
4
2x  3 ≥ 8
2x ≥ 8 + 3
2x ≥ 5
1
x ≥ 2
2

The smallest integer satisfying the inequality
2x  3
≥ 2 is –2.
4
-- ans end --
13043023
Find the smallest integer satisfying the inequality
5x  4
7.
7
A. 9
B. 10
C. 11
D. 12
-- ans -Solution:
The answer is B.
5x  4
 7
7
5x + 4 > 49
5x > 45
x > 9

The smallest integer satisfying the inequality
5x  4
 7 is 10.
7
-- ans end -P. 12
13043024
Find the greatest integer satisfying the inequality 3(x + 2) ≥ 7x  4.
A. 3
B. 2
C. 1
D. 1
-- ans -Solution:
The answer is B.
3(x + 2)
3x + 6
10
1
2
2
7x  4
≥ 7x  4
≥ 4x
≥
≥
x
1
2
 The greatest integer satisfying the inequality 3(x + 2) ≥ 7x  4 is 2.
-- ans end --
i.e.
x ≤ 2
13043025
In the solutions of the inequality
5  2x
≤ 3, how many of them are positive integers smaller than
3
3?
A. 0
B. 1
C. 2
D. 3
-- ans -Solution:
The answer is C.
5  2x
≤ 3
3
5  2x
2x
2x
x
≤
9
≤ 95
≤ 4
≥ 2
P. 13
5  2x
≤ 3.
3

1 and 2 satisfy the inequality

There are two positive integers smaller than 3, which satisfy the inequality
5  2x
≤ 3.
3
-- ans end --
13043026
The sum of two consecutive even numbers is smaller than 76. Find the range of values of the
greater number (let it be x).
A. x > 37
B.
C.
D.
x < 37
x > 39
x < 39
-- ans -Solution:
The answer is D.
 The greater number is x,
the smaller number is x  2.
According to the given information,

(x  2) + x < 76
2x  2 < 76
2x < 78

x < 39
-- ans end --
13043027
The sum of the first three multiples of a number is smaller than 120. Find the greatest value of the
greatest multiple.
A. 19
B. 20
C. 57
D. 60
-- ans -Solution:
The answer is C.
Let the greatest multiple be x,
P. 14
x
2x
and
.
3
3
According to the given information,
x 2x

 x  120
3 3
6x
 120
3
2 x  120
then the remaining multiples are
x  60

The greatest integer that is smaller than 60 and is divisible by 3 is 57.
 The greatest multiple is 57.
-- ans end --
13043028
The length and the width of a rectangle are (7 + a) cm and (5 + a) cm respectively. If
its perimeter is not less than 40 cm, what is the minimum area of the rectangle?
A. 4 cm2
B. 55 cm2
C. 99 cm2
D. 399 cm2
-- ans -Solution:
The answer is C.
According to the given information,
2(7 + a + 5 + a) ≥ 40
12 + 2a ≥ 20
2a ≥ 8
a ≥ 4
The minimum area of the rectangle = (7 + 4)×(5 + 4) cm2
= 99 cm2
-- ans end --
13043029
The revision time of Rosan in the past four days were 70 min, 50 min, 85 min and
x min respectively. If her average revision time in the past four days was more than an hour, find the
smallest value of x (in integer).
A. 34
B. 35
P. 15
C.
D.
36
37
-- ans -Solution:
The answer is C.
70  50  85  x
> 60
4
205 + x > 240
x > 35
 The smallest value of x is 36.
-- ans end -13043030
In a Mathematics competition, every correct answer awards 10 marks, 5 marks will be deducted for
every wrong answer and no marks will be deducted for not answering the question. Given that
Raymond has answered 20 questions, and his score was not less than 100. At least how many
questions did Raymond answer correctly?
A. 6
B. 7
C. 13
D. 14
-- ans -Solution:
The answer is D.
Let Raymond answered x questions correctly,
then he answered (20  x) questions wrongly.
According to the given information,
10x – 5(20 – x) ≥ 100
10x – 100 + 5x ≥ 100
15x ≥ 200
1
x ≥ 13
3
 He answered at least 14 questions correctly.
-- ans end --
13043031
P. 16
The percentages of the ingredients of fruit juices A and B are shown in the following table.
Ingredient
Fruit juice A
Fruit juice B
Water
20%
10%
Orange juice
30%
70%
Lemon juice
50%
20%
1.5 L of fruit juice A is mixed with y L of fruit juice B. Find the range of the value of y so that the
mixed fruit juice contains more than 45% of orange juice.
A. y > 1
B. y < 1
C. y > 0.9
D. y < 0.9
-- ans -Solution:
The answer is C.
According to the given information,
1.5  30%  y  70%
100%  45%
1.5  y
0.45 + 0.7y > 0.45(1.5 + y)
0.7y  0.45y > 0.675  0.45
0.25y > 0.225
y > 0.9
-- ans end --
13043032
In a bag, there are 20 yellow balls and several green balls. It is known that the number
of yellow balls is less than three times that of the green balls by at most 3. At
most how many green balls are there inside the bag?
A. 5
B. 6
C. 7
D. 8
-- ans -Solution:
The answer is C.
Let x be the number of green balls inside the bag.
According to the given information,
P. 17
3x  20 ≤ 3
3x ≤ 23
2
x ≤ 7
3
 There are at most 7 green balls inside the bag.
-- ans end ---End--
P. 18