Linear Inequalities – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Linear inequalities compare expressions using <, >, ≤ or ≥. Their solutions are intervals on a number line or half-planes in two variables.

Rules

• Adding or subtracting the same number keeps the inequality sign.
• Multiplying or dividing by a positive number keeps the sign; by a negative number reverses it.
• Solutions are written as intervals or shown on a number line.

Two variables

• ax + by < c represents a half-plane; the boundary line is dashed for < or > and solid for ≤ or ≥.
• Test a point such as (0, 0) to choose the correct side.
• A system of inequalities has its solution in the common shaded region.

Key terms

Half-plane

region on one side of a line

Solution set

all values satisfying the inequality

Closed interval

[a, b] includes both end points; shown with filled dots on a number line

Important questions with answers

Quick revision

• Multiply by negative ⇒ flip sign.
• Dashed line for strict inequality.
• Test (0, 0) for the side.

Exam tips

• −2x > 6 ⇒ x < −3 (sign flips when dividing by −2).
• When solving word problems, translate "at least" as ≥ and "at most" as ≤.

Practice MCQs

1. Solution of x + 3 > 5 is

   1. x > 2
   2. x < 2
   3. x > 8
   4. x = 2
   Show answer

   Answer: (a) x > 2. Subtract 3.

2. Multiplying an inequality by a negative number

   1. keeps the sign
   2. has no effect
   3. makes it an equation
   4. reverses the sign
   Show answer

   Answer: (d) reverses the sign. Rule.

3. Solution of −3x ≤ 12 is

   1. x ≤ −4
   2. x ≥ −4
   3. x ≥ 4
   4. x ≤ 4
   Show answer

   Answer: (b) x ≥ −4. Divide by −3 and flip.

4. The point (0, 0) satisfies

   1. x + y > 2
   2. x + y < 2
   3. x − y > 1
   4. 2x + 3y ≥ 6
   Show answer

   Answer: (b) x + y < 2. 0 < 2.

5. Solution of 3(x − 1) ≤ 2(x + 3) is

   1. x ≤ 9
   2. x ≥ 9
   3. x ≤ 3
   4. x ≤ −9
   Show answer

   Answer: (a) x ≤ 9. 3x − 3 ≤ 2x + 6.

6. Integer solutions of 2 < x + 1 ≤ 5 are

   1. 3, 4
   2. 1, 2, 3, 4
   3. 2, 3, 4, 5
   4. 2, 3, 4
   Show answer

   Answer: (d) 2, 3, 4. 1 < x ≤ 4.

7. The boundary of x + 2y ≥ 4 is drawn as

   1. a dashed line
   2. a solid line
   3. no line
   4. a curve
   Show answer

   Answer: (b) a solid line. Includes equality.

8. Solution set of −5 ≤ (2 − 3x)/4 ≤ 9 is

   1. [−11, 7]
   2. [−22/3, 34/3]
   3. (−34/3, 22/3)
   4. [−34/3, 22/3]
   Show answer

   Answer: (d) [−34/3, 22/3]. −20 ≤ 2 − 3x ≤ 36.