When both sides of an inequation are multiplied by −2, the inequality sign:

Is reversed. Multiplying by a negative number reverses the sign.

The solution of −2x ≥ 6 is:

x ≤ −3. Divide by −2 and reverse the sign.

The solution of |x| < 3 is:

−3 < x < 3. |x| < a ⇔ −a < x < a.

The solution of |x| ≥ 2 is:

x ≤ −2 or x ≥ 2. |x| ≥ a ⇔ x ≤ −a or x ≥ a.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 8: Linear Inequations

Short notes, key concepts, quick revision and practice MCQs with answers

An inequation compares two expressions using <, >, ≤ or ≥. The chapter solves linear inequations in one variable and shows solutions on the number line, graphs linear inequations in two variables as half-planes and finds the common solution region of a system of inequations, a foundation for linear programming.

Key concepts: Linear Inequations

Inequation: A statement involving real numbers or algebraic expressions with the symbols <, >, ≤ or ≥, e.g. 3x − 5 < 7.
Rules for solving: Adding or subtracting the same number on both sides keeps the sign. Multiplying or dividing by a positive number keeps the sign; by a negative number reverses it.
One variable: The solution set is an interval, shown on the number line: a hollow circle for an excluded end point (<, >) and a filled circle for an included one (≤, ≥).
Modulus inequalities: For a > 0: |x| < a ⇔ −a < x < a; |x| ≤ a ⇔ −a ≤ x ≤ a; |x| > a ⇔ x < −a or x > a; |x| ≥ a ⇔ x ≤ −a or x ≥ a.
Sign rules: ab > 0 or a/b > 0 means a and b have the same sign; ab < 0 or a/b < 0 means opposite signs.
Two variables: ax + by ≤ c represents a half-plane. Draw the line ax + by = c (solid for ≤, ≥; dotted for <, >), test a point such as the origin and shade the side that satisfies the inequation.
System of inequations: The solution of several inequations is the region common to all their half-planes. x ≥ 0, y ≥ 0 restrict the region to the first quadrant.

Quick revision: important points

Multiplying by −1 reverses the inequality sign.
If the origin satisfies the inequation, shade the side containing the origin.
x ≥ 0 is the region on and to the right of the Y-axis; y ≥ 0 is on and above the X-axis.
A double inequality a ≤ px + q < b is solved by operating on all three parts.

Practice MCQs with answers

When both sides of an inequation are multiplied by −2, the inequality sign:
1. Remains the same
2. Is reversed
3. Becomes =
4. Is removed
Show answer
Answer: B. Is reversed
Multiplying by a negative number reverses the sign.
The solution of 3x − 5 < 7 is:
1. x < 4
2. x > 4
3. x < 2/3
4. x ≤ 4
Show answer
Answer: A. x < 4
3x < 12 gives x < 4.
The solution of −2x ≥ 6 is:
1. x ≥ −3
2. x ≤ −3
3. x ≥ 3
4. x ≤ 3
Show answer
Answer: B. x ≤ −3
Divide by −2 and reverse the sign.
The solution of |x| < 3 is:
1. x < 3
2. −3 < x < 3
3. x > −3
4. x < −3 or x > 3
Show answer
Answer: B. −3 < x < 3
|x| < a ⇔ −a < x < a.
The solution of |x| ≥ 2 is:
1. −2 ≤ x ≤ 2
2. x ≤ −2 or x ≥ 2
3. x ≥ 2
4. x ≤ 2
Show answer
Answer: B. x ≤ −2 or x ≥ 2
|x| ≥ a ⇔ x ≤ −a or x ≥ a.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters