9.5 Solving linear inequalities — Learn

Prerequisites: solving equations and representing inequalities on a number line.

Objectives: solve linear inequalities, preserve the inequality direction under addition, subtraction and positive division, and reverse it when multiplying or dividing by a negative.

This section covers solving linear inequalities in one variable, within AQA reference A22.

Solve an inequality using the balance method. Adding or subtracting the same value keeps its direction. Multiplying or dividing by a negative reverses the direction because the order of numbers reverses.

Example 1. Solve $3x+2\leq14$.

Subtract $2$ from both sides: $3x\leq12$.

Divide by positive $3$: $x\leq4$.

Example 2. Solve $-2x<8$.

Divide both sides by $-2$ and reverse the sign: $x>-4$.

Check with $x=0$: $0<8$, so the solution direction is plausible.

Watch out. Dividing by a negative without reversing the sign gives the wrong side. Since $-2<1$, dividing both sides by $-1$ gives $2>-1$.

Show the final solution on a number line when asked. Use the boundary style that matches strict or inclusive inequality.

You should now be able to

• solve a linear inequality using inverse operations;
• reverse the inequality when multiplying or dividing by a negative;
• check and represent a solution set.