Arpita Srivastava Content Writer
Content Writer | Updated On - Jul 24, 2025
Linear Inequalities are a form of inequality that is used to express a linear function. It is a form of algebraic expression that compares polynomials of degree 1.
- Linear Inequalities is an important topic included in NCERT Class 11 Mathematics.
- It determines non-equal comparison between two numbers or expressions.
- ‘<','>', '≤' or '≥'" are the symbols used in linear inequalities.
- The expressions using the symbol ‘>' or '<' are known as strict inequalities.
- The expressions using the symbol ‘≤' or '≥' are known as slack inequalities.
- These inequalities are considered similar to linear equations.
- Linear equations replace the inequality sign with the equality sign.
- Polynomial, rational, and absolute value inequalities are three types of linear inequalities.
- It denotes a mathematical expression which can be created using one or two variables.
- The concept can be used when we are planning a party, so it helps in planning the party budget.
The four operations performed on linear inequalities are as follows:
- Addition of linear inequalities is given as:
a > y, then a + b > y + b
- Subtraction of linear inequalities is given as:
a > y, then a - b > y - b
- Multiplication of linear inequalities is given as:
a > y, then a x b > y x b
- Division of inequalities is given as:
a > y, then a / b > y / b
Linear Inequalities MCQs
Ques: The length of a rectangle is four times the breadth. If the minimum perimeter of the rectangle is 160 cm, then
- breadth > 16 cm
- length < 20 cm
- breadth x ≥ 16 cm
- length ≤ 20 cm
Click here for the answer
Ans: (c) breadth x ≥ 16 cm
Explanation: Let x be the breadth of a rectangle.
⇒ As it is given that length = 4x
= Given that the minimum perimeter of a rectangle is 160 cm.
= 2 (4x + x) ≥ 160
= 5x ≥ 80
Therefore the result is x ≥ 16
Ques: Consider a linear inequality where – 3x + 15 < – 12, then
- x ∈ (9, ∞)
- x ∈ [9, ∞)
- x ∈ (– ∞, 9]
- x ∈ [– 9, 10)
Click here for the answer
Ans: (a) x ∈ (9, ∞)
Explanation: Given, -3x + 15 < -12
⇒ Subtracting 15 from both sides,
= -3x + 15 – 15 < -12 – 15
= -3x < -27
⇒ x > 9 {since the division by negative number inverts the inequality sign}
Therefore the result is x ∈ (9, ∞)
Ques: Suppose we have three real numbers which include x, y and b are real and it is given that x < y, b < 0. Determine the relation between three variables.
- x/b < y
- x ≤ y
- x/b < y/b
- x ≥ y/b
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Ans: (c) x/b < y/b
Explanation: Since it is given that x, y and b are real numbers where x < y, b < 0.
⇒ Suppose, x < y
⇒ Divide both sides of the inequality by “b”
Therefore x/b < y/b {as b < 0}
Ques: If |x −2| > 6, then
- x ∈ (– 4, 8)
- x ∈ [– 4, 8]
- x ∈ (– ∞, – 4) U (8, ∞)
- x ∈ [– ∞, – 4) U [8, ∞)
Click here for the answer
Ans: (c) x ∈ (– ∞, – 4) ∪ (8, ∞)
Explanation: |x – 2| > 6
⇒ x – 2 < – 6 and x – 2 > 6
⇒ x < -4 and x > 8
Therefore, x ∈ (-∞, -4) U (8, ∞)
Ques: If |x – 8|/(x – 8) ≥ 0, then
- x ∈ [8, ∞)
- x ∈ (7, 8)
- x ∈ (– ∞, 8)
- x ∈ (8, ∞)
Click here for the answer
Ans: (d) x ∈ (8, ∞)
Explanation: Since it is given that: |x – 8|/(x – 8) ≥ 0
⇒ This is possible when x − 8 ≥ 0, and x – 8 ≠ 0.
⇒ Here, x ≥ 8 but x ≠ 8
Therefore, x > 8, i.e. x ∈ (8, ∞).
Ques: If |x + 5| ≥ 10, then
- x ∈ (– 15, 15]
- x ∈ (– 15, 5]
- x ∈ (– ∞, – 15] ∪ [5, ∞)
- x ∈ [– ∞, – 15] ∪ [5, ∞)
Click here for the answer
Ans: (d) x ∈ (– ∞, – 15] ∪ [5, ∞)
Explanation: Since it is given that, |x + 5| ≥ 10
⇒ x + 5 ≤ – 10 or x + 5 ≥ 10
⇒ x ≤ – 15 or x ≥ 5
Therefore the result is x ∈ (– ∞, – 15] ∪ [5, ∞)
Ques: If 4x + 4 < 6x +5, then x belongs to the interval
- (2, ∞)
- (-1/2, ∞)
- (-∞, 1/2)
- (-4, ∞)
Click here for the answer
Ans: (b) (-1/2, ∞)
Explanation: Since it is given that, 4x + 4 < 6x + 5
⇒ Subtracting 4 from both sides,
= 4x + 4 – 4 < 6x + 5 – 4
= 4x < 6x + 1
⇒ Subtracting 6x from both sides,
= 4x – 6x < 6x + 1 – 6x
⇒ – 2x < 1 or
⇒ x > – 1/2 i.e., all the real numbers greater than –1/2, are the solutions of the given inequality.
Hence, the solution set is (–1/2, ∞), i.e. x ∈ (-1/2, ∞)
Ques: Solve the linear inequalities in this linear inequality 2x - 17 > 3 - 8x
- x > =2
- x < 2
- x <= 2
- x > 2
Click here for the answer
Ans: (d) x > 2
Explanation: Since it is given that: 2x - 5 > 3 - 8x
⇒ 2x + 8x > 3 + 17
⇒ 10x > 20
Therefore the result is x > 2
Ques: Isha scored 80 and 80 marks in the first two-class test. Calculate the minimum marks he should get in the third test to have an average of at least 90 marks.
- x ≥ 90
- x > 90
- x <= 90
- x < 90
Click here for the answer
Ans: (a) x ≥ 90
Explanation: Suppose that x is the mark obtained by Ravi in the third unit test.
⇒ It is given that the student should have an average of at least 60 marks.
⇒ From the given information, we can write the linear inequality as
⇒ (80 + 80 + x)/3 ≥ 90
⇒ Now, simplify the expression: (160 + x) ≥ 270
⇒ x ≥ 270 -160
Therefore the result is x ≥ 90
Ques: What is the formula used for addition of two linear inequalities:
- a + b > y + b
- a > y
- a - b > y - b
- a x b > y x b
Click here for the answer
Ans: (a) a + b > y + b
Explanation: The addition of two linear inequalities are as follows: a + b > y + b
Ques: What is the formula used for subtraction of two linear inequalities:
- a + b > y + b
- a > y
- a - b > y - b
- a x b > y x b
Click here for the answer
Ans: (c) a - b > y - b
Explanation: The subtraction of two linear inequalities are as follows: a + b > y + b
Ques: Calculate 5x + 2 > 12 when x is a real number.
- (-2, 2)
- (-1, ∞)
- (-2, ∞)
- (-2, 1)
Click here for the answer
Ans: (c)(-2, ∞)
Explanation: 5x + 2 > 12
⇒ 5x > 10
⇒ x > -2 ……Dividing by 5
Hence the solution (-2, ∞)
Ques: When x is an integer. Calculate the value of x through linear inequalities: 5x – 2 ≤ 3x + 2
- {..., -2, -1, 0, 1}
- {...,-3, -2, -1, 0, 1}
- {..., -2, -1, 0, 1,2}
- {..., -2, -1, 0, 1,2,3}
Click here for the answer
Ans: (a) {..., -2, -1, 0, 1}
Explanation: 5x-2 ≤ 3x+2
⇒ 5x-2x ≤ 1+2
⇒ 3x ≤ 3
⇒ x ≤ 1
Hence the solution {..., -2, -1, 0, 1}
Ques: 50x < 150, where x is the natural number then determine x:
- {1,2,3,4}
- {1,2}
- {0,1,2}
- {1,2,3}
Click here for the answer
Ans: (b) {1,2}
Explanation: It is given that 50x < 150
⇒ Dividing both sides by 50
⇒ x < 3
Hence the solution set {1,2}
Ques: Solve if 9x > -36
- -4
- -3
- -5
- 4
Click here for the answer
Ans:(a) -4
Explanation: It is given that 9x > -36
⇒ x > -4 ………(Divide by 6)
Therefore the result is x > -4
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