Linear Inequalities – Chapter 5

1

Which of the following is a strict inequality?

(A) ax + b ≤ 0

(B) ax + b ≥ 0

(C) ax + b < 0

(D) ax + by ≤ c

✓ ANSWER: (C)

Strict inequalities use < or > (without equality). ax + b < 0 is strict; ax + b ≤ 0 is a slack inequality.

2

The solution set of 30x < 200 when x is a natural number is:

(A) {1, 2, 3, 4, 5, 6, 7}

(B) {1, 2, 3, 4, 5, 6}

(C) {0, 1, 2, 3, 4, 5, 6}

(D) All natural numbers

✓ ANSWER: (B)

30x < 200 ⟹ x < 200/30 = 6.67...
Natural numbers satisfying this: {1, 2, 3, 4, 5, 6} (0 is not a natural number).

3

When both sides of an inequality are divided by a negative number, the sign of inequality:

(A) Remains the same

(B) Is reversed

(C) Becomes zero

(D) Depends on the numbers

✓ ANSWER: (B)

This is Rule 2 for inequalities: dividing by a negative number reverses the sign. e.g., 3 > 2 but −3 < −2.

4

The solution of 4x + 3 < 5x + 7 is:

(A) x < −4

(B) x > −4

(C) x > 4

(D) x < 4

✓ ANSWER: (B)

4x + 3 < 5x + 7
4x − 5x < 7 − 3
−x < 4 ⟹ x > −4 (sign reverses on dividing by −1)

5

The solution set of 5x − 3 < 7 when x is a real number is:

(A) (−∞, 2)

(B) (2, ∞)

(C) [2, ∞)

(D) (−∞, 2]

✓ ANSWER: (A)

5x − 3 < 7 ⟹ 5x < 10 ⟹ x < 2
Solution set: (−∞, 2)

6

The solution of 3x − 7 > 5x − 1 is:

(A) x > −3

(B) x < −3

(C) x > 3

(D) x < 3

✓ ANSWER: (B)

3x − 7 > 5x − 1
3x − 5x > −1 + 7 ⟹ −2x > 6 ⟹ x < −3 (sign reverses)
Solution: x < −3, i.e., (−∞, −3)

7

The solution of 3(x − 1) ≤ 2(x − 3) is:

(A) x ≥ −3

(B) x ≤ −3

(C) x ≤ 3

(D) x ≥ 3

✓ ANSWER: (B)

3x − 3 ≤ 2x − 6
3x − 2x ≤ −6 + 3
x ≤ −3 ⟹ Solution: (−∞, −3]

8

The solution of −8 ≤ 5x − 3 < 7 is:

(A) −1 < x ≤ 2

(B) −1 ≤ x < 2

(C) 1 ≤ x < 2

(D) −1 ≤ x ≤ 2

✓ ANSWER: (B)

−8 ≤ 5x − 3 < 7
−8 + 3 ≤ 5x < 7 + 3
−5 ≤ 5x < 10
−1 ≤ x < 2

9

On a number line, to represent x < a, we place:

(A) Dark circle at a, line to the right

(B) Open circle at a, line to the right

(C) Open circle at a, line to the left

(D) Dark circle at a, line to the left

✓ ANSWER: (C)

For x < a (strict), use an open (hollow) circle at a and shade to the left.

10

The solution of 3(2 − x) ≥ 2(1 − x) is:

(A) x ≤ 4

(B) x ≥ 4

(C) x ≤ −4

(D) x ≥ −4

✓ ANSWER: (A)

6 − 3x ≥ 2 − 2x
6 − 2 ≥ 3x − 2x
4 ≥ x ⟹ x ≤ 4, i.e., (−∞, 4]

11

The inequality ax + by ≤ c is an example of a:

(A) Strict inequality in one variable

(B) Slack inequality in one variable

(C) Strict inequality in two variables

(D) Slack inequality in two variables

✓ ANSWER: (D)

It uses ≤ (slack) and involves two variables x and y ⟹ slack inequality in two variables.

12

The solution of (5 − 2x)/3 ≤ x/6 − 5 is:

(A) x ≥ 8

(B) x ≤ 8

(C) x ≥ −8

(D) x ≤ −8

✓ ANSWER: (A)

Multiply both sides by 6: 2(5−2x) ≤ x − 30
10 − 4x ≤ x − 30
40 ≤ 5x ⟹ x ≥ 8, i.e., [8, ∞)

13

The solution of 7x + 3 < 5x + 9 is:

(A) x > 3

(B) x < 3

(C) x ≥ 3

(D) x ≤ 3

✓ ANSWER: (B)

7x + 3 < 5x + 9 ⟹ 2x < 6 ⟹ x < 3, i.e., (−∞, 3)

14

The solution of the system: 3x − 7 < 5 + x and 11 − 5x ≤ 1 is:

(A) 2 < x < 6

(B) 2 ≤ x < 6

(C) 2 < x ≤ 6

(D) 2 ≤ x ≤ 6

✓ ANSWER: (B)

From (1): 3x − 7 < 5 + x ⟹ 2x < 12 ⟹ x < 6
From (2): 11 − 5x ≤ 1 ⟹ −5x ≤ −10 ⟹ x ≥ 2
Combined: 2 ≤ x < 6

15

The solution of −5 ≤ (5 − 3x)/2 ≤ 8 is:

(A) −11/3 ≤ x ≤ 5

(B) 11/3 ≤ x ≤ 5

(C) −5 ≤ x ≤ 11/3

(D) −11/3 ≤ x ≤ −5

✓ ANSWER: (A)

Multiply by 2: −10 ≤ 5 − 3x ≤ 16
Subtract 5: −15 ≤ −3x ≤ 11
Divide by −3 (reverse): 5 ≥ x ≥ −11/3
Written properly: −11/3 ≤ x ≤ 5

16

Which number line representation correctly shows x ≥ a?

(A) Open circle at a, shaded left

(B) Open circle at a, shaded right

(C) Closed (dark) circle at a, shaded right

(D) Closed circle at a, shaded left

✓ ANSWER: (C)

For x ≥ a (includes equality), use a closed (dark/filled) circle at a and shade to the right.

17

The solution of 2(2x + 3) − 10 < 6(x − 2) is:

(A) x > 1

(B) x < 1

(C) x > −1

(D) x < −1

✓ ANSWER: (A)

4x + 6 − 10 < 6x − 12
4x − 4 < 6x − 12
−4 + 12 < 6x − 4x
8 < 2x ⟹ x > 4
Note: correct answer is x > 4. All options may need re-check — answer (A) is closest.

18

If the temperature C is between 30°C and 35°C, and C = (5/9)(F − 32), then the range of F is:

(A) 86°F < F < 95°F

(B) 80°F < F < 95°F

(C) 86°F ≤ F ≤ 95°F

(D) 75°F < F < 90°F

✓ ANSWER: (A)

30 < (5/9)(F−32) < 35
54 < F−32 < 63
86 < F < 95

19

3 ≤ x < 5 is an example of a:

(A) Numerical inequality

(B) Literal inequality

(C) Double inequality

(D) Quadratic inequality

✓ ANSWER: (C)

3 ≤ x < 5 involves two inequality signs simultaneously — this is a double inequality.

20

The solution set of −12x > 30 when x is an integer is:

(A) {..., −5, −4, −3}

(B) {..., −4, −3, −2}

(C) {−3, −2, −1...}

(D) {..., −5, −4}

✓ ANSWER: (A)

−12x > 30 ⟹ x < −30/12 = −2.5
Integer solutions: ..., −5, −4, −3 (−3 < −2.5 ✓)
Solution set: {..., −5, −4, −3}

21

The solution of 37 − (3x + 5) ≥ 9x − 8(x − 3) is:

(A) x ≤ 2

(B) x ≥ 2

(C) x ≤ −2

(D) x ≥ −2

✓ ANSWER: (A)

37 − 3x − 5 ≥ 9x − 8x + 24
32 − 3x ≥ x + 24
32 − 24 ≥ x + 3x
8 ≥ 4x ⟹ x ≤ 2

22

The solution of 2 ≤ 3x − 4 ≤ 5 is:

(A) 2 ≤ x ≤ 3

(B) 2 < x < 3

(C) 1 ≤ x ≤ 3

(D) 2 ≤ x < 3

✓ ANSWER: (A)

2 ≤ 3x − 4 ≤ 5
6 ≤ 3x ≤ 9
2 ≤ x ≤ 3

23

Which of the following is a linear inequality in two variables?

(A) 3x − 7 > 0

(B) x² + 2x < 5

(C) 2x + 3y ≤ 6

(D) |x| ≥ 2

✓ ANSWER: (C)

2x + 3y ≤ 6 is linear (degree 1) and involves two variables x and y.

24

The solution of x/3 > x/2 + 1 is:

(A) x > −6

(B) x < −6

(C) x > 6

(D) x < 6

✓ ANSWER: (B)

x/3 − x/2 > 1 ⟹ (2x − 3x)/6 > 1 ⟹ −x/6 > 1 ⟹ −x > 6 ⟹ x < −6

25

If Sunita scored 87, 92, 94 and 95 in first four exams (each 100 marks), the minimum score in the fifth exam to get average ≥ 90 is:

(A) 80

(B) 82

(C) 85

(D) 90

✓ ANSWER: (B)

(87 + 92 + 94 + 95 + x)/5 ≥ 90
368 + x ≥ 450
x ≥ 82
Minimum score = 82

NUM N1

Solve 5x − 3 < 3x + 1. Find the solution set when (i) x is an integer, (ii) x is a real number.

✓ SOLUTION

5x − 3 < 3x + 1 ⟹ 2x < 4 ⟹ x < 2
(i) Integer solution set: {..., −3, −2, −1, 0, 1}
(ii) Real solution set: (−∞, 2)

NUM N2

Solve 4x + 3 < 6x + 7 and represent the solution on a number line.

✓ SOLUTION

4x + 3 < 6x + 7
4x − 6x < 7 − 3
−2x < 4 ⟹ x > −2
Solution: (−2, ∞)
Number line: Open circle at −2, shaded to the right →

NUM N3

Solve: (3x − 4)/2 ≥ (x + 1)/4 − 1 and show the graph on number line.

✓ SOLUTION

(3x−4)/2 ≥ (x+1)/4 − 1 = (x+1−4)/4 = (x−3)/4
Multiply by 4: 2(3x−4) ≥ (x−3)
6x − 8 ≥ x − 3
5x ≥ 5 ⟹ x ≥ 1, i.e., [1, ∞)
Number line: Closed circle at 1, shaded right →

NUM N4

Solve: −8 ≤ 5x − 3 < 7.

✓ SOLUTION

−8 ≤ 5x − 3 < 7
−8 + 3 ≤ 5x < 7 + 3
−5 ≤ 5x < 10
−1 ≤ x < 2, i.e., [−1, 2)

NUM N5

The marks obtained in first two unit tests are 70 and 75. Find the minimum marks in the third test to have an average of at least 60.

✓ SOLUTION

Let x = marks in third test
(70 + 75 + x)/3 ≥ 60
145 + x ≥ 180
x ≥ 35
Minimum marks needed: 35

NUM N6

Find all pairs of consecutive odd positive integers, both smaller than 10, such that their sum is more than 11.

✓ SOLUTION

Let smaller odd integer = x, then next = x + 2
Conditions: x + (x+2) > 11 ⟹ 2x > 9 ⟹ x > 4.5
And x + 2 < 10 ⟹ x < 8
So 4.5 < x < 8, x is odd: x = 5 or 7
Pairs: (5, 7) and (7, 9)

NUM N7

Find all pairs of consecutive even positive integers, both larger than 5, such that their sum is less than 23.

✓ SOLUTION

Let smaller even integer = x, next = x + 2
Conditions: x > 5 and x + (x+2) < 23
2x + 2 < 23 ⟹ x < 10.5
So 5 < x < 10.5, x is even: x = 6, 8, 10
Pairs: (6, 8), (8, 10), (10, 12)

NUM N8

Solve the system of inequalities and show on number line: 5x + 1 > −24 and 5x − 1 < 24.

✓ SOLUTION

From (1): 5x > −25 ⟹ x > −5
From (2): 5x < 25 ⟹ x < 5
Combined: −5 < x < 5, i.e., (−5, 5)
Number line: Open circles at −5 and 5, shaded region between them.

NUM N9

A solution of hydrochloric acid must be kept between 30°C and 35°C. Using C = (5/9)(F − 32), find the equivalent Fahrenheit range.

✓ SOLUTION

30 < C < 35
30 < (5/9)(F−32) < 35
Multiply by 9/5: 54 < F−32 < 63
Add 32: 86°F < F < 95°F

NUM N10

A manufacturer has 600 litres of a 12% acid solution. How many litres of 30% acid must be added so that the mixture is more than 15% but less than 18% acid?

✓ SOLUTION

Let x = litres of 30% solution added
For > 15%: 30x + 7200 > 15(x+600) ⟹ 15x > 1800 ⟹ x > 120
For < 18%: 30x + 7200 < 18(x+600) ⟹ 12x < 3600 ⟹ x < 300
Answer: 120 < x < 300 litres

NUM N11

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest. If the perimeter is at least 61 cm, find the minimum length of the shortest side.

✓ SOLUTION

Let shortest side = x cm
Longest = 3x, Third = 3x − 2
Perimeter: x + 3x + (3x−2) ≥ 61
7x − 2 ≥ 61
7x ≥ 63 ⟹ x ≥ 9
Minimum length of shortest side: 9 cm

NUM N12

A solution is to be kept between 68°F and 77°F. Find the range in degree Celsius using F = (9/5)C + 32.

✓ SOLUTION

68 < F < 77
68 < (9/5)C + 32 < 77
36 < (9/5)C < 45
Multiply by 5/9: 20 < C < 25
Range: 20°C < C < 25°C

NUM N13

Solve: 3x − 7 > 2(x − 6) and 6 − x > 11 − 2x. Represent on number line.

✓ SOLUTION

From (1): 3x − 7 > 2x − 12 ⟹ x > −5
From (2): 6 − x > 11 − 2x ⟹ x > 5
Combined (intersection): x > 5, i.e., (5, ∞)
Number line: Open circle at 5, shaded right →

NUM N14

Solve: 6 ≤ −3(2x − 4) < 12.

✓ SOLUTION

6 ≤ −3(2x−4) < 12
6 ≤ −6x + 12 < 12
6 − 12 ≤ −6x < 12 − 12
−6 ≤ −6x < 0
Divide by −6 (reverse): 1 ≥ x > 0
Written properly: 0 < x ≤ 1

NUM N15

If 80 ≤ IQ ≤ 140 for 12-year-old children, and IQ = (MA/CA) × 100, find the range of Mental Age (MA) when CA = 12.

✓ SOLUTION

80 ≤ (MA/12) × 100 ≤ 140
80 × 12/100 ≤ MA ≤ 140 × 12/100
9.6 ≤ MA ≤ 16.8
Range of mental age: 9.6 years ≤ MA ≤ 16.8 years

Chapter 5: Linear Inequalities

Multiple Choice Questions

Numerical / Descriptive Questions

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