Linear Inequations

ICSE Class 10 Mathematics • Chapter 03

1. Introduction

Inequation: A mathematical statement that uses inequality symbols (>, <, ≥, ≤) instead of equals sign.

Linear Inequation: Inequation where the highest power of variable is 1.

Symbol Meaning Example
> Greater than x > 5
<< /td> Less than x < 3
≥ Greater than or equal to x ≥ 2
≤ Less than or equal to x ≤ 7

2. Properties of Inequations

Rule 1: Adding/subtracting same number on both sides doesn't change the inequality.

If $a > b$, then $a + c > b + c$ and $a - c > b - c$

Rule 2: Multiplying/dividing by a POSITIVE number keeps inequality unchanged.

If $a > b$ and $c > 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$

Rule 3 (CRITICAL!): Multiplying/dividing by a NEGATIVE number REVERSES the inequality.

If $a > b$ and $c < 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$

Golden Rule: When you multiply or divide by a negative number, FLIP THE SIGN!

$>$ becomes $<$ and $<$ becomes $>$

$≥$ becomes $≤$ and $≤$ becomes $≥$

3. Number Sets (Replacement Sets)

Set Symbol Members
Natural Numbers N {1, 2, 3, 4, ...}
Whole Numbers W {0, 1, 2, 3, ...}
Integers Z {..., -2, -1, 0, 1, 2, ...}
Real Numbers R All rational and irrational numbers

4. Solving Linear Inequations

Steps:
  1. Simplify both sides (remove brackets, combine like terms)
  2. Move variable terms to one side, constants to other
  3. Divide by coefficient (remember to flip sign if negative!)
  4. Write solution set based on replacement set

Example 1: Solve $3x - 5 < 7$, where $x \in N$

$3x - 5 < 7$

$3x < 12$

$x < 4$

Solution Set in N: {1, 2, 3}

Example 2: Solve $-2x + 3 ≥ 9$, where $x \in Z$

$-2x ≥ 6$

$x ≤ -3$ (sign flipped because dividing by -2)

Solution Set in Z: {..., -5, -4, -3}

Example 3: Solve $\frac{2x-1}{3} ≥ \frac{3x-2}{4} - 1$, where $x \in R$

Multiply by LCM (12):

$4(2x-1) ≥ 3(3x-2) - 12$

$8x - 4 ≥ 9x - 6 - 12$

$8x - 4 ≥ 9x - 18$

$-4 + 18 ≥ 9x - 8x$

$14 ≥ x$ or $x ≤ 14$

Solution: $\{x : x ≤ 14, x \in R\}$

5. Double Inequations (Continued Inequations)

Form: $a < x < b$ or $a ≤ x ≤ b$

This means x is between a and b.

Method: Solve as one unit, performing same operation on all three parts.

Example 4: Solve $-3 ≤ 2x - 1 < 5$, where $x \in Z$

Add 1 to all parts: $-2 ≤ 2x < 6$

Divide by 2: $-1 ≤ x < 3$

Solution Set in Z: {-1, 0, 1, 2}

6. Graphical Representation

Inequality On Number Line
$x > a$ Open circle at a, arrow pointing right →
$x < a$ Open circle at a, arrow pointing left ←
$x ≥ a$ Filled circle at a, arrow pointing right →
$x ≤ a$ Filled circle at a, arrow pointing left ←
$a < x < b$ Open circles at a and b, line between
$a ≤ x ≤ b$ Filled circles at a and b, line between
[Number line: x ≥ 2 shown with filled circle at 2 and arrow to right]

7. Quick Reference

Operation Effect on Inequality
Add/Subtract any number No change
Multiply/Divide by positive No change
Multiply/Divide by negative REVERSE the sign

Exam Practice (PYQ Trends)

PYQ: 2023

BOARD Solve and graph: $-2 ≤ \frac{1}{2} - \frac{2x}{3} ≤ 1\frac{5}{6}$, $x \in N$

PYQ: 2022

BOARD Given $15 - 7x > 2x - 27$. If $x \in N$, list the solution set and represent on number line.