Inequalities

Video lesson4 worked examples

Theory

Inequalities (or inequations) are solved using the exact same steps and inverse operations as linear equations, but with an inequality sign in the middle.

When evaluating real-life situations, limitations should be considered (e.g., the maximum safe load for a concrete beam).

The Golden Rule: solve an inequality exactly like an equation — except that multiplying or dividing both sides by a negative number flips the direction of the sign. A neat way to avoid this altogether is to collect the letters on whichever side keeps their coefficient positive.

⚠️ Common Examiner Traps

Forgetting to flip: dividing by a negative reverses the sign — $-3y \geq -6$ becomes $y \leq 2$ , not $y \geq 2$ .
Flipping when you add or subtract: the sign only flips for multiplying or dividing by a negative, never for adding or subtracting.
Fractions and brackets: clear denominators and expand brackets exactly as for an equation before isolating the variable.
Answer is a range: the solution is an inequality such as $x > 4$ , not a single value.

Worked examples

Example 1

Solving a standard Inequation

Solve $4x + 7 < 23$ .

Step 1: Subtract 7 from both sides: $4x < 16$ .

Answer: $x < 4$ .

Example 2

Negative Variables (sign flip)

Solve $12 - 3y \geq 6$ .

Step 1: Subtract 12 from both sides: $-3y \geq -6$ .

Step 2: Divide by −3, remembering to flip the inequality sign. Answer: $y \leq 2$ .

Example 3

With Brackets

Solve $3(x - 2) < x + 8$ .

Step 1: Expand the bracket: $3x - 6 < x + 8$ .

Step 2: Subtract $x$ from both sides and add 6 to both: $2x < 14$ .

Step 3: Divide by 2 (positive, so no flip). Answer: $x < 7$ .

Example 4

🎯 Exam-style

Solve the inequation $\dfrac{x + 2}{3} \leq \dfrac{x}{2}$ .

Step 1: Clear the denominators by multiplying every term by the LCM, 6:

2(x + 2) \leq 3x

Step 2: Expand: $2x + 4 \leq 3x$ .

Step 3: Subtract $2x$ from both sides: $4 \leq x$ .

Step 4: Read it the usual way round. Answer: $x \geq 4$ .

Ready to practise?

15 Linear Inequalities questions with answers and video solutions