Linear Inequations • Topic 1 of 2

Solution of Linear Inequations

What is a linear inequation? A linear inequation is a mathematical statement that compares two linear expressions using inequality symbols. The solution of a linear inequation is the set of all values of the variable that satisfy the inequality.

Inequality symbols and their meanings:

Symbol	Meaning	Example
<	less than	x < 5
>	greater than	x > 5
≤	less than or equal to	x ≤ 5
≥	greater than or equal to	x ≥ 5

Rules for solving linear inequations:

Adding or subtracting the same number from both sides does NOT change the inequality
Multiplying or dividing both sides by a positive number does NOT change the inequality
Multiplying or dividing both sides by a negative number REVERSES the inequality sign

Important: When you multiply or divide by a negative number, flip the inequality sign!

Example: −2x < 6 → Divide by −2 (negative!) → x > −3 (sign flips from < to >)

Real-life analogy: Think of an inequation like a see-saw. Adding weight to both sides keeps the balance (same direction). But if you multiply by a negative, it's like flipping the see-saw upside down — everything reverses!

┌─────────────────────────────────────────────────────────────┐
│         SOLVING LINEAR INEQUATIONS - STEP-BY-STEP            │
└─────────────────────────────────────────────────────────────┘

BASIC STEPS:

    Start: 2x + 3 < 11
          │
          ▼ Subtract 3 from both sides
    2x + 3 - 3 < 11 - 3
          │
          ▼ Simplify
    2x < 8
          │
          ▼ Divide both sides by 2 (positive, sign stays same)
    x < 4
    
    Solution: x is any number less than 4


WHEN SIGN FLIPS (Multiplying/Dividing by Negative):

    Start: -3x + 5 ≥ 14
          │
          ▼ Subtract 5 from both sides
    -3x ≥ 9
          │
          ▼ Divide by -3 (NEGATIVE — FLIP SIGN!)
    x ≤ -3
    
    Solution: x is any number less than or equal to -3


SOLVING WITH VARIABLES ON BOTH SIDES:

    Start: 5x - 7 > 3x + 5
          │
          ▼ Subtract 3x from both sides
    2x - 7 > 5
          │
          ▼ Add 7 to both sides
    2x > 12
          │
          ▼ Divide by 2 (positive)
    x > 6


RULES COMPARISON TABLE:

┌─────────────────────────────────────────────────────────────┐
│  Operation              │  Effect on Inequality Sign        │
├─────────────────────────┼───────────────────────────────────┤
│  Add/Subtract same number│  No change (sign stays same)      │
│  Multiply/Divide by +ve  │  No change                         │
│  Multiply/Divide by -ve  │  REVERSES the sign (< becomes >)  │
└─────────────────────────┴───────────────────────────────────┘


COMMON MISTAKES TO AVOID:

    ✗ Forgetting to flip sign when dividing by negative
    ✗ Only flipping one side of the inequality
    ✗ Treating ≤ like < when graphing
    ✗ Not simplifying fully before solving

Solved Examples

Worked Example

Solve: 4x − 7 > 9

Solution

Step 1: Add 7 to both sides: 4x − 7 + 7 > 9 + 7
Step 2: Simplify: 4x > 16
Step 3: Divide both sides by 4 (positive): x > 4

Answer: x > 4

Worked Example

Solve: 3(x − 2) ≤ 5x + 4

Solution

Step 1: Expand left side: 3x − 6 ≤ 5x + 4
Step 2: Subtract 3x from both sides: −6 ≤ 2x + 4
Step 3: Subtract 4 from both sides: −10 ≤ 2x
Step 4: Divide both sides by 2: −5 ≤ x, which is the same as x ≥ −5

Answer: x ≥ −5

Worked Example

Solve: 2 − 5x ≥ 3x − 14

Solution

Step 1: Add 5x to both sides: 2 ≥ 8x − 14
Step 2: Add 14 to both sides: 16 ≥ 8x
Step 3: Divide by 8: 2 ≥ x, which is x ≤ 2

Answer: x ≤ 2

Key Points

Linear inequations use <, >, ≤, or ≥ symbols
Adding/subtracting same number from both sides does not change inequality
Multiplying/dividing by a positive number does not change inequality
Multiplying/dividing by a negative number reverses the inequality sign
Always simplify both sides before isolating the variable
The solution set contains all values that satisfy the inequality

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Multiplying or dividing an inequation by a negative number:

Explanation: The inequality sign flips.

Q2.The solution of $2x>6$ is:

Explanation: Divide by $2$.

Q3.The set from which solutions are taken is called the:

Explanation: The replacement set restricts solutions.

Q4.If $x\in\mathbb{N}$ and $x<4$, the solution set is:

Explanation: Natural numbers below $4$.

Practice more MCQs → 📝 Assignment · 35 marks