Year 10: Solving linear inequalities

Inequalities are like equations with attitude: instead of declaring two sides equal, they tell you that one side is bigger or smaller than the other. Mastering them unlocks everything from speed‑limit word problems to shading feasible regions in senior maths.

1. What is a linear inequality?

A linear inequality looks like a linear equation but uses one of four symbols:

$<$ “less than”
$\leq$ “less than or equal to”
$>$ “greater than”
$\geq$ “greater than or equal to”

Example: $5 x - 3 > 7$ .

2. Solve it like an equation—almost

You can add, subtract, multiply, or divide both sides just as you do with equations except for one critical twist: whenever you multiply or divide by a negative number, you must flip the inequality sign.

Example:

 $5 x - 3 > 7 + 3 + 3 (add 3) 5 x > 10 \div 5 \div 5 (divide b y + 5, sign stays the same) x > 2$

Exercise

3. Flipping the sign with negatives

Watch the direction change:

Example:

 $- 2 x + 8 \leq 0 - 2 x \leq - 8 (subtract 8) \div (- 2) \div (- 2) (divide b y - 2 \to flip \leq to \geq) x \geq 4$

4. Showing the answer on a number line

Graphing helps you “see” the solution set:

Open circle for $<$ or $>$ (value not included).
Closed circle for $\leq$ or $\geq$ (value included).
Shade to the right for “greater than”, left for “less than”.

Exercise

5. Compound inequalities

Sometimes a variable is trapped between two numbers, e.g. $3 < 2 x + 1 \leq 9$ . Treat it as two linked inequalities:

Example:

 $3 < 2 x + 1 \leq 9 - 1 - 1 - 12 < 2 x \leq 8 \div 2 \div 2 \div 21 < x \leq 4$

The solution is every number strictly greater than 1 and up to and including 4.

6. Checking your solution

Pick a convenient value from the shaded region and substitute it back. For $x > 2$ , choose $x = 3$ :

 $5 (3) - 3 = 12 which is indeed > 7 ✔$

7. Common mistakes

Forgetting to flip the sign: Always change $<$ to $>$ (or vice‑versa) when multiplying or dividing by a negative.
Misreading open vs closed circles: Double‑check whether the boundary value counts.
Dropping the variable’s coefficient sign: Keep track of negatives before deciding whether to flip.
Testing the wrong region: When unsure, test a point on each side of the boundary to confirm shading direction.

8. Why inequalities matter

Inequalities power everything from designing safe load limits in engineering to coding “if” statements in software. By learning to solve and graph them now, you’re equipping yourself with a toolkit that stretches far beyond the classroom.

So next time you face a linear inequality, remember: treat it like an equation, flip the sign for negatives, graph it clearly, and test a point. Confidence—and correct answers—will follow!

Practice!

Here are some exercise quizzes to further sharpen your skills in solving inequalities:

Solve inequalities in a single variable