GCSE Maths · Algebra

Inequalities

Linear inequalities, representing solutions on number lines, and (Higher) quadratic and graphical inequalities.

200 GCSE-style practice questions

Learn

Inequalities, explained point by point

Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.

  1. Use inequality notation and show inequalities on a number line

    x > 2 means anything bigger than 2; x ≥ 2 includes 2 itself. On a number line, an open circle means the end value is NOT included (< or >) and a filled circle means it is (≤ or ≥), with an arrow or bar showing the covered region.

  2. Solve linear inequalities in one variable

    Solve exactly like an equation, keeping the inequality sign: 3x + 2 < 11 gives x < 3. The one exception: multiplying or dividing by a NEGATIVE number flips the sign - from −2x < 6 you get x > −3. Test a value in the original to check.

  3. List the integer solutions of an inequality

    Solve first, then read off the whole numbers. For −3 < x ≤ 2 the integers are −2, −1, 0, 1, 2 - the strict < excludes −3, the ≤ includes 2. Double-inequalities like −5 ≤ 2x + 1 < 7 are solved on all three parts at once.

  4. Solve quadratic inequalitiesHigher only

    Solve the matching equation first, then sketch the parabola to see where it is above or below zero. x² − x − 6 < 0 has roots −2 and 3; the U-shape dips below the axis BETWEEN them, so −2 < x < 3. "Greater than" picks the two outside regions instead.

  5. Show inequalities as regions on a graphHigher only

    Draw each boundary line (solid for ≤/≥, dashed for </>) and shade according to the inequality - test a point like (0, 0) if unsure which side. With several inequalities, the answer is the region satisfying all of them at once; label it R.

Practice

Try a Inequalities question

A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.

Solve 4x − 3 > 2x + 7.

  1. x > 2
  2. x < 5
  3. x > 5
  4. x ≥ 5
Show the answer and the working

Answer: x > 5

Collect the x terms on one side: 2x > 10, so x > 5.

  1. Subtract 2x from both sides: 2x − 3 > 7.
  2. Add 3 to both sides: 2x > 10.
  3. Divide both sides by 2: x > 5.

Make Inequalities stick

A free account adds worked examples from our question bank, from a warm-up to a stretch, plus ask-Wiz - a coach you can put any point from this lesson to - and a checklist that remembers which points you've studied. No card needed.

Continue learning free

Ready to practise Inequalities?

Start your free trial to unlock this topic - questions that adapt to how you're doing, with instant encouraging feedback on every answer. No card needed.

Start practising free

Already have an account? Log in

More Algebra topics

Keep reading