Skip to content
MathAnvil
§ Algebra

Inequalities

CCSS.7.EECCSS.HSA.REI3 min read

Inequalities form the backbone of algebraic thinking in Years 8-11, bridging the gap between simple equations and complex mathematical modelling. When Charlotte spends less than £15 at the tuck shop or Harry needs more than 60% to pass his GCSE, they're using inequality concepts without realising it.

Try it right now

Click “Generate a problem” to see a fresh example of this technique.

§ 01

Why it matters

Inequalities appear everywhere in real-world problem-solving, from calculating minimum scores needed for university entry to determining budget constraints for school trips. A Year 11 student might need at least 75 marks across 5 GCSE papers to achieve their target grade, creating the inequality 5x ≥ 375, where x represents the average score needed per paper. Business studies teachers use inequalities when exploring profit margins—if a school canteen needs to make at least £200 profit daily and each meal generates £3.50 profit, they need to sell at least 58 meals (3.5x ≥ 200). Construction projects rely on inequalities for material calculations, whilst sports coaches use them to set training targets. Understanding inequalities develops logical reasoning skills essential for GCSE Foundation and Higher tier success, particularly in problem-solving contexts where students must interpret 'at least', 'no more than', and 'between' language.

§ 02

How to solve inequalities

Inequalities

  • Solve like an equation (same operations on both sides).
  • If you multiply or divide by a negative, FLIP the sign.
  • Graph on a number line (open circle for < >, closed for ≤ ≥).

Example: -2x > 6 → x < -3 (sign flipped).

§ 03

Worked examples

Beginner§ 01

x + 3 < 8

Answer: x < 5

  1. Understand the problem → x + 3 < 8 — This is like an equation, but instead of '=' we have '<'. We solve it the same way.
  2. Subtract 3 from both sides → x + 3 − 3 < 8 − 3 → x < 5 — Isolate x by removing the constant from the left side.
  3. Check with a test value → Try x = 4: 4 + 3 = 7 < 8 ✓ — Pick a value of x that satisfies x < 5 and verify it works in the original inequality.
Easy§ 02

2x + 4 ≥ 20

Answer: x ≥ 8

  1. Write the inequality → 2x + 4 ≥ 20 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
  2. Subtract 4 from both sides → 2x + 4 − 4 ≥ 20 − 4 → 2x ≥ 16 — Remove the constant term from the left side. The inequality sign stays the same.
  3. Divide both sides by 2 → x ≥ 8 — Divide by 2 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
  4. Verify with a test value → Try x = 9: 2·9 + 4 = 18 + 4 = 22 ≥ 20? ✓ — Pick x = 9 (which satisfies x ≥ 8) and check it works in the original inequality.
Medium§ 03

8x + 3 ≤ -53

Answer: x ≤ -7

  1. Write the inequality → 8x + 3 ≤ -53 — Our goal is to isolate x, just like solving an equation — but watch out when dividing by a negative number!
  2. Subtract 3 from both sides → 8x + 3 − 3 ≤ -53 − 3 → 8x ≤ -56 — Remove the constant term from the left side. The inequality sign stays the same.
  3. Divide both sides by 8 → x ≤ -7 — Divide by 8 to isolate x. The inequality sign stays the same since we're dividing by a positive number.
  4. Verify with a test value → Try x = -8: 8·-8 + 3 = -64 + 3 = -61 ≤ -53? ✓ — Pick x = -8 (which satisfies x ≤ -7) and check it works in the original inequality.
§ 04

Common mistakes

  • Forgetting to flip the inequality sign when dividing by negative numbers, leading to -2x > 6 becoming x > -3 instead of x < -3
  • Confusing open and closed circles on number lines, drawing a closed circle for x < 5 instead of an open circle
  • Mixing up 'less than' and 'greater than' symbols, writing x > 8 when the solution should be x < 8
  • Not checking solutions by substituting test values, missing errors like writing x ≥ 10 when x ≤ 10 is correct
Practice on your own
Generate unlimited inequality practice problems tailored to your Year 8 or GCSE classes with MathAnvil's free worksheet generator.
Generate free worksheets
§ 05

Frequently asked questions

Why do we flip the inequality sign when multiplying or dividing by negative numbers?
When you multiply both sides of an inequality by a negative number, you reverse the relationship. Think of temperatures: if 5°C > -2°C, then multiplying by -1 gives -5°C < 2°C. The colder temperature becomes the smaller value, so the inequality flips to maintain the true relationship.
How do I know whether to use an open or closed circle on a number line?
Use a closed circle (filled in) for ≤ and ≥ because the boundary value is included in the solution set. Use an open circle (not filled) for < and > because the boundary value is not part of the solution.
What's the difference between solving inequalities and equations in Year 8?
The process is nearly identical—isolate the variable using inverse operations. The key difference is remembering to flip the inequality sign when multiplying or dividing by negative numbers. Equations have one solution; inequalities typically have infinitely many solutions within a range.
How do quadratic inequalities work at GCSE level?
Quadratic inequalities like x² - 5x + 6 < 0 require factorising first, then using a sign chart or graphical method to determine where the expression is positive or negative. The solution often involves ranges like 2 < x < 3.
Why do some inequality problems have 'no solution' or 'all real numbers' as answers?
This happens when simplifying leads to impossible statements (like 5 < 2, meaning no solution) or always-true statements (like -3 < 7, meaning all real numbers work). These special cases often appear in GCSE examination questions.
§ 06

Related topics

Share this article