Solving Inequalities

Solving inequalities is a core algebra skill in Edexcel IGCSE Maths. This page shows you how to solve linear and compound inequalities, when to reverse the inequality sign after multiplying or dividing by a negative, and how to show a solution on a number line using open and closed circles. Work through the step-by-step worked examples, then use the free, randomly generated, auto-marked practice questions below to build speed and accuracy for the exam.

Prior Knowledge Before starting, be confident with Solving Linear Equations and Simplifying Expressions (collecting like terms). You solve an inequality in the same way as an equation, so those skills come first.
⇩ Jump to Practice Questions ⇩

Solving Inequalities: the Core Ideas

Solve like an equation Isolate the variable one step at a time using inverse operations. Collect the terms in \(x\) on one side and the numbers on the other.
Reverse the sign for negatives When you multiply or divide both sides by a negative number, flip the inequality: \(\lt\) becomes \(\gt\), and \(\le\) becomes \(\ge\).
Show the solution set Draw a number line: an open circle for \(\lt\) or \(\gt\), a filled circle for \(\le\) or \(\ge\). List the integer solutions if the question asks.

Step-by-step method

  1. Collect like terms: move the \(x\) terms to one side and the numbers to the other.
  2. Isolate \(x\): divide or multiply to leave a single \(x\). If you divide or multiply by a negative, reverse the inequality sign.
  3. Write the solution in inequality form, for example \(x \gt 3\) or \(-2 \lt x \le 4\).
  4. Draw a number line: open circle for a strict inequality, filled circle for "or equal to".
  5. List the integers in the range if a solution set is asked for.
The sign-flip rule

Multiplying or dividing both sides by a negative number reverses the inequality sign. Dividing by \(-2\) turns \(\gt\) into \(\lt\). This is the single most common mistake in the topic. If you would rather avoid it, move the negative \(x\) term to the other side first (see Example 2), so you only ever divide by a positive.

Open circle: strict inequality (\(\lt\) or \(\gt\)); the endpoint is not included Filled circle: \(\le\) or \(\ge\); the endpoint is included

Worked Examples

💡 Example 1: two-step linear

Solve \(\;3x + 4 \ge 19\) and show the result on a number line.

\[ \begin{array}{rcl} 3x + 4 &\ge& 19 \\ 3x &\ge& 15 \\ x &\ge& 5 \end{array} \]
What is happening?
  • Subtract 4 from both sides.
  • Divide both sides by 3 (positive, so no flip).
  • Filled circle at 5 because of "or equal to".
1 2 3 4 5 6 7 8 9

\(x \ge 5\): filled circle at 5, shading to the right

💡 Example 2: reverse the sign

Solve \(\;7 - 2x \lt 1\) and show the result on a number line.

\[ \begin{array}{rcl} 7 - 2x &\lt& 1 \\ -2x &\lt& -6 \\ x &\gt& 3 \end{array} \]
What is happening?
  • Subtract 7 from both sides.
  • Divide by \(-2\): the sign flips from \(\lt\) to \(\gt\).
  • Open circle at 3 because it is strict.
0 1 2 3 4 5 6 7 8

\(x \gt 3\): open circle at 3, shading to the right

💡 Example 3: brackets and both sides

Solve \(\;5(x - 1) \ge 2x + 7\).

\[ \begin{array}{rcl} 5(x-1) &\ge& 2x + 7 \\ 5x - 5 &\ge& 2x + 7 \\ 3x &\ge& 12 \\ x &\ge& 4 \end{array} \]
What is happening?
  • Expand the bracket first.
  • Subtract \(2x\) and add 5 to both sides.
  • Divide by 3 (positive, no flip).
0 1 2 3 4 5 6 7 8

\(x \ge 4\): filled circle at 4, shading to the right

💡 Example 4: compound inequality

Solve \(\;-5 \lt 2x - 1 \le 7\) and list the integer solutions.

\[ \begin{array}{ccccc} -5 &\lt& 2x - 1 &\le& 7 \\ -4 &\lt& 2x &\le& 8 \\ -2 &\lt& x &\le& 4 \end{array} \]
What is happening?
  • Do the same step to all three parts.
  • Add 1, then divide by 2.
  • Open at \(-2\), filled at 4.
-3 -2 -1 0 1 2 3 4 5

\(-2 \lt x \le 4\): integers are \(-1, 0, 1, 2, 3, 4\)

🔑 Key Points

  • Solve step by step like an equation: isolate \(x\) with inverse operations.
  • Flip the sign only when multiplying or dividing by a negative number.
  • \(x \gt 4\) means \(x\) cannot equal 4; \(x \ge 4\) means \(x\) can equal 4 or be greater.
  • Open circle for \(\lt, \gt\); filled circle for \(\le, \ge\).
  • For a compound inequality, do the same step to all three parts.

⚠️ Common Pitfalls

  • Forgetting to reverse the sign when dividing by a negative.
  • Using the wrong circle (open vs filled) on the number line.
  • Not expanding brackets or collecting terms before dividing.
  • Listing integers outside the range, or missing an endpoint that is included.

✍️ Exam Tip

Write one step per line and keep the inequality sign in every line. If a coefficient of \(x\) is negative, either divide by the negative and flip the sign, or move that term across so you divide by a positive. When asked to show the answer, always draw a number line and substitute a test value back into the original inequality to check.

⇩ Jump to Practice Questions ⇩

This page covers linear and compound inequalities. Next, learn to solve quadratic inequalities with a parabola, or how to shade regions on a graph.

Next: Quadratic Inequalities →

Solving Inequalities: Practice Rooms

Randomly generated, auto-marked practice in solving inequalities for Edexcel IGCSE Maths. The rooms rise in difficulty: solve linear inequalities (Room 1), reverse the sign when you multiply or divide by a negative (Room 2), solve double-sided compound inequalities (Room 3), list the integer solutions (Room 4), read and choose number lines (Room 5), then a mixed set (Room 6). Every grid gives 16 fresh questions and each correct answer adds to your global streak.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Type an inequality with the keyboard or the < ≤ > ≥ buttons; write <= for ≤ and >= for ≥, and use a slash for a fraction, e.g. x>=2/3. Room 3 gives two boxes, one for each part of the range (either order). Room 4 wants every integer solution, separated by commas. Answers mark when you click away.