Year 10 - Linear inequalities - Free practice

Year 10 Linear Inequalities Worksheets

Q: What's the key rule for inequalities?

If you multiply or divide both sides by a negative number, you must flip the inequality sign. So -2x > 4 becomes x < -2 (note the flip). Forgetting this is the single most common Year 10 inequality mistake.

Q: How do students represent inequality solutions on a number line?

Open circle for strict ( ), filled circle for inclusive ( =). Arrow extending in the direction of the solution. For x > 3, open circle at 3, arrow pointing right.

Solve, graph on number line, handle sign flips from negatives.

Quick answer

How do you solve linear inequalities?

Solve a linear inequality like an equation, keeping the inequality sign throughout, but flip the sign whenever you multiply or divide both sides by a negative number. For example, solving 2x + 3 > 7 gives x > 2. Graph the solution on a number line using an open circle for strict inequalities and a closed circle for ≤ or ≥.

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Worked examples

Year 10 linear inequalities: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Solving a one-variable linear inequality
Solve $3x - 4 > 11$ and represent the solution on a number line.
Show worked solution
Key idea: Solve a linear inequality the same way as a linear equation, keeping the inequality sign pointing the same direction (as long as you only add or subtract, or multiply/divide by a positive number).
1. Add 4 to both sides: $3x - 4 + 4 > 11 + 4 \implies 3x > 15$
2. Divide both sides by 3 (positive, so the sign stays): $x > 5$
3. On a number line, draw an open circle at 5 (since $x = 5$ is not included) and shade the ray going to the right.
Answer: $x > 5$
Example 2 · Inequality with a fraction and sign flip
Solve $\dfrac{-2x}{3} + 1 \leq 5$ and represent the solution on a number line.
Show worked solution
Key idea: When you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip direction.
1. Subtract 1 from both sides: $\dfrac{-2x}{3} \leq 4$
2. Multiply both sides by 3 (positive, sign stays): $-2x \leq 12$
3. Divide both sides by $-2$ (negative, so flip the sign): $x \geq -6$
4. On a number line, draw a closed circle at $-6$ (since $x = -6$ is included) and shade the ray going to the right.
Answer: $x \geq -6$
Example 3 · Inequality in context: saving for a purchase
Anika has already saved $85. She earns $12 per hour working at a cafe and wants to save enough to buy a jacket that costs at least $181. Write and solve an inequality to find the minimum number of whole hours she must work.
Show worked solution
Key idea: Translate the real-life situation into an inequality, solve it, then interpret the answer in context.
1. Let $h$ be the number of hours Anika works. Her total savings will be $85 + 12h$ dollars. She needs this to be at least $181: $85 + 12h \geq 181$
2. Subtract 85 from both sides: $12h \geq 96$
3. Divide both sides by 12 (positive, sign stays): $h \geq 8$
4. Since $h$ must be a whole number and $h \geq 8$ , the minimum number of hours is 8.
Answer: Anika must work at least 8 hours.

Common mistakes

Where students usually lose marks

Forgetting to flip the inequality sign

$-3x > 12 \implies x > -4$ (sign not flipped when dividing by $-3$ )

Whenever you multiply or divide both sides by a negative number, the inequality sign must reverse direction. Here the correct result is $x < -4$ . A good habit is to circle the sign as a reminder to flip it whenever you use a negative multiplier or divisor.

Using a closed circle instead of an open circle (or vice versa)

Drawing a closed circle at 5 for $x > 5$ , suggesting $x = 5$ is a solution.

Use an open circle when the endpoint is not included (strict inequalities $<$ and $>$ ) and a closed circle when it is included ( $\leq$ and $\geq$ ). Ask yourself: does the boundary value actually satisfy the inequality?

Treating the inequality like an equation and losing the sign

$2x - 3 < 7 \implies 2x - 3 = 7 \implies x = 5$ , then writing $x < 5$ without checking.

While the algebraic steps are similar, always carry the inequality symbol through every line of working. Write $<$ (or whichever sign applies) in every step, not just at the end, so you can spot if a sign flip is needed.

Multiplying only one term when clearing a fraction

$\dfrac{x-2}{3} > 4 \implies x - 2 > 4$ (multiplied 3 into the numerator but forgot to multiply the right-hand side by 3)

When you multiply both sides by a number to clear a fraction, every term on both sides must be multiplied. The correct step is $\dfrac{x-2}{3} \times 3 > 4 \times 3$ , giving $x - 2 > 12$ .

Try one yourself

Your turn

Solve the inequality $3(2x - 1) > 4x + 7$ and find the smallest integer that satisfies it.

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About this worksheet

Why we built it

Year 10 inequalities mostly mirror equations - with one crucial twist: dividing by a negative flips the sign. These worksheets give your student varied multi-step inequalities and number-line representations to hammer that rule home.

What's covered

Sub-skills your student will practise

✓Solving one-variable linear inequalities
✓Inequalities with fractions
✓Inequalities with brackets and like-term collection
✓Sign-flipping when multiplying / dividing by a negative
✓Representing the solution on a number line

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Frequently asked

Questions parents ask about linear inequalities

What's an inequality?

A statement that two expressions are not equal but in a specific relationship - greater than (>), less than (<), greater than or equal to (>=), less than or equal to (<=). Solving them gives a range of values, not a single number.

What's the key rule for inequalities?

If you multiply or divide both sides by a negative number, you must flip the inequality sign. So -2x > 4 becomes x < -2 (note the flip). Forgetting this is the single most common Year 10 inequality mistake.

How do students represent inequality solutions on a number line?

Open circle for strict (< or >), filled circle for inclusive (<= or >=). Arrow extending in the direction of the solution. For x > 3, open circle at 3, arrow pointing right.

Are quadratic inequalities a Year 10 topic?

Mainstream Year 10 stays with linear inequalities. Quadratic inequalities (where x^2 < 9 or similar) arrive in Year 11 Methods. The technique - sketch the parabola, identify where it's below/above zero - is a Methods skill.

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