# Inequalities

> Edexcel International GCSE Mathematics A · 4MA1 (2016)
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-inequalities/

This guide teaches you to solve linear and (for Higher tier) quadratic inequalities, represent solutions on number lines, and shade/identify inequality regions on Cartesian graphs, aligned to Edexcel IGCSE Maths A 4MA1 spec 2.8.

**Prerequisites:** [Solving linear equations](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-linear-equations/); [Plotting straight line graphs](https://www.owlsprep.com/study/edexcel-igcse-math-a-s3-straight-line-graphs/); [(Higher only) Solving quadratic equations](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-quadratic-equations/)

## Learning objectives

- Interpret and use inequality symbols (<, >, ≤, ≥) including double-ended inequalities
- Represent linear inequality solutions on number lines using open/closed circle conventions
- Solve linear inequalities in one variable, including cases with negative coefficients
- Shade and identify regions defined by linear inequalities on Cartesian graphs
- (Higher only) Solve quadratic inequalities in one variable and represent solutions on number lines
- (Higher only) Identify complex regions defined by multiple linear inequalities

## Inequality Notation & Number Line Representation

**Inequality Symbols** — The four core inequality symbols are: < (less than, strict), > (greater than, strict), ≤ (less than or equal to, non-strict), ≥ (greater than or equal to, non-strict). Double-ended inequalities combine two conditions, e.g. $2 < x \leq 7$ means x is greater than 2 and less than or equal to 7.

When representing inequalities on a number line, use an open circle for strict inequalities (boundary value not included) and a closed filled circle for non-strict inequalities (boundary value included).

**Worked example:** Represent the inequality $-1 \leq x < 3$ on a number line.

1. Identify the boundary values: $x = -1$ and $x = 3$.
2. For $x=-1$, the inequality uses $\leq$, so draw a closed filled circle at -1.
3. For $x=3$, the inequality uses $<$, so draw an open circle at 3.
4. Draw a solid line connecting the two circles to show all values between them are included in the solution set.

> **Exam tip:** Always label your number line clearly with boundary values and whole number markers. Marks are awarded for both correct line placement and correct circle type, so double check these before moving on.

## Solving Linear Inequalities

Solving linear inequalities follows almost identical rules to solving linear equations, with one critical exception: if you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality sign.

> **warning**
>
> Forgetting to reverse the inequality sign when multiplying/dividing by a negative number is one of the most common mark-losing mistakes on this topic. Always check if you are using a negative multiplier/divisor before finalising your solution.

**Worked example:** Solve $5 - 2x \geq 9$ and represent the solution on a number line.

1. Subtract 5 from both sides: $-2x \geq 4$
2. Divide both sides by -2, and reverse the inequality sign: $x \leq -2$
3. Represent on a number line: closed circle at -2, with a solid line extending left to show all values less than or equal to -2 are included.

**Worked example:** Solve the double-ended inequality $4 < 2x + 6 \leq 12$.

1. Subtract 6 from all three parts of the inequality: $4-6 < 2x \leq 12-6$ which simplifies to $-2 < 2x \leq 6$
2. Divide all three parts by 2 (positive, so no sign reversal): $-1 < x \leq 3$

*Calculator:* allowed

## Inequalities on Cartesian Graphs

To represent a linear inequality on a Cartesian graph, first draw the boundary line for the corresponding linear equation. For strict inequalities (<, >) the boundary line is dashed, while for non-strict inequalities it is solid (Edexcel IGCSE does not require boundary convention marking for Foundation tier, but consistency is recommended). Then shade the region that satisfies all given inequalities.

**Worked example:** Shade the region that satisfies the three inequalities: $x \geq 0$, $y \geq x$, $x + y \leq 4$. Label the region R.

1. Draw boundary line $x=0$ (the y-axis): this is a solid line, shade to the right of the line.
2. Draw boundary line $y=x$: this is a solid line, shade above the line (test point (0,2): $2 \geq 0$ which is true).
3. Draw boundary line $x+y=4$ (rearranged to $y=-x+4$): this is a solid line, shade below the line (test point (0,0): $0+0 \leq 4$ which is true).
4. The overlapping shaded region satisfies all three inequalities: label this region R.

> **Exam tip:** Always test a point not on the boundary line (e.g. (0,0) if it is not on the line) to confirm which side of the line satisfies the inequality. This eliminates mistakes from misremembering shading rules. For Higher tier, you may be given more complex boundary lines like $y=2x+1$ or $5x+2y=20$, use the exact same method.

## (Higher Only) Solving Quadratic Inequalities

**Critical Values** — The roots of the corresponding quadratic equation for an inequality, which form the boundaries of the solution set.

For quadratic inequalities with a positive $x^2$ coefficient (upward opening parabola): if the inequality is > 0, the solution is the set of values outside the two critical values; if it is < 0, the solution is the set of values between the two critical values. Always test a value if you are unsure of the region.

**Worked example:** Solve $x^2 - 3x - 10 < 0$ and represent the solution on a number line.

1. Find critical values by solving the corresponding quadratic equation: $x^2 - 3x -10 = 0$
2. $$(x-5)(x+2) = 0 \implies x=5, x=-2$$
3. The inequality is < 0, and the $x^2$ coefficient is positive, so the solution is between the two roots: $-2 < x < 5$
4. Represent on a number line: open circles at -2 and 5, connected by a solid line to show all values between them are included.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to reverse the inequality sign when multiplying/dividing by a negative number
  - Why it fails: Multiplying/dividing by a negative reverses the order of values, so the inequality direction must flip to keep the statement true
  - Correct: Whenever you multiply or divide both sides of an inequality by a negative number, immediately flip < to >, ≤ to ≥, or vice versa.
- **Wrong:** Using the wrong circle type on number lines (closed for strict, open for non-strict)
  - Why it fails: This incorrectly shows whether the boundary value is included in the solution set, costing easy method marks
  - Correct: Use open circles for <, >; closed filled circles for ≤, ≥. Double check circle types before moving to the next question.
- **Wrong:** Shading the wrong region on Cartesian graphs
  - Why it fails: You mixed up the side of the line that satisfies the inequality, leading to an incorrect region
  - Correct: Always test a point not on the boundary line (e.g. (0,0) if it is not on the line) to confirm which side of the line satisfies the inequality.
- **Wrong:** (Higher only) Choosing the wrong region for quadratic inequalities (e.g. picking between roots for $x^2 > a$)
  - Why it fails: You forgot the shape of the parabola, or did not test a value to confirm the region
  - Correct: For positive $x^2$ coefficient: > 0 = solution outside roots, < 0 = solution between roots. Test a value in the region to confirm if unsure.
- **Wrong:** Splitting double-ended inequalities into two separate inequalities unnecessarily, leading to arithmetic errors
  - Why it fails: Unnecessary extra steps increase the chance of calculation mistakes and sign errors
  - Correct: Apply operations (add, subtract, multiply, divide) to all three parts of the double-ended inequality at the same time to keep calculations simple.

## Cheatsheet

| Task | Foundation Tier Rule | Higher Tier Addendum |
| --- | --- | --- |
| Interpret symbols | <: open circle, >: open circle, ≤: closed circle, ≥: closed circle | Same as foundation |
| Solve linear inequalities | Same steps as linear equations, reverse sign if ×/÷ by negative | Same as foundation |
| Number line representation | Connect correct circle types with solid line for solution set | Same for linear; open circles for strict quadratic inequalities |
| Graph shading | Draw boundary lines, shade overlapping region, label with R | Handle complex linear boundaries (e.g. 5x+2y=20) using same method |
| Solve quadratic inequalities | Not required | Find critical values, positive $x^2$: >0 = outside roots, <0 = between roots |

## What's next

Now that you have mastered inequalities for Edexcel IGCSE Maths A, you can apply these skills to tackle more complex algebra topics and exam-style questions. Inequalities are often combined with other topics like sequences, graphs, and optimisation problems in Higher tier papers, so practicing mixed-topic questions will help you prepare for full-mark responses. Make sure you complete both Foundation and Higher tier past paper questions to test your understanding of all conventions and problem types for this topic.

- [Quadratic Equations (Higher Tier)](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-quadratic-equations/)

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