Simple linear and quadratic inequalities

Edexcel International GCSE Further Pure Mathematics· 3D· 15 min read

1. Solving Simple Linear Inequalities★★☆☆☆⏱ 4 min

✓ Calculator OK

📘 Definition

Linear inequality

An inequality that can be written in the form $a x + b ⋈ c x + d$ , where $⋈$ is $<, >, \leq$ or $\geq$ , and the highest power of $x$ is 1.

To solve linear inequalities, follow the same steps as rearranging linear equations, with one key rule: reverse the inequality sign if you multiply or divide both sides by a negative number.

📐 Worked Example

Solve the inequality $3 x + 7 \geq 19 - 2 x$ , giving your answer in the form $x ⋈ k$ .

1
Add $2 x$ to both sides to collect $x$ terms on the left: $5 x + 7 \geq 19$
2
Subtract 7 from both sides: $5 x \geq 12$
3
Divide by 5 (positive, no sign flip required): $x \geq \frac{12}{5} = 2.4$

📐 Worked Example

Solve the inequality $11 - 4 x < 3$ .

1
Subtract 11 from both sides: $- 4 x < - 8$
2
Divide by -4 (negative, flip sign): $x > 2$

Exam tip:

Always double-check if you multiplied/divided by a negative number at the final step — forgetting to flip the inequality sign is the most common linear inequality mistake.

2. Rearranging Quadratic Inequalities & Finding Critical Values★★★☆☆⏱ 5 min

✓ Calculator OK

Quadratic inequalities are rearranged first to get all terms on one side, with 0 on the other, and a positive leading coefficient to simplify sign checking. Critical values are the roots of the corresponding quadratic equation.

📘 Definition

Standard form of quadratic inequality

A quadratic inequality written as $a x^{2} + b x + c ⋈ 0$ , where $a > 0$ , and $⋈$ is $<, >, \leq$ or $\geq$ .

📐 Worked Example

Rearrange $2 x^{2} + 3 x - 5 < x^{2} + 2 x + 1$ into standard form and find its critical values.

1
Subtract the right-hand side from both sides: $x^{2} + x - 6 < 0$
2
Factorise the quadratic expression: $(x + 3) (x - 2) = 0$
3
Solve for $x$ to get critical values: $x = - 3$ and $x = 2$

3. Selecting Solution Regions for Quadratic Inequalities★★★☆☆⏱ 6 min

✓ Calculator OK

Once you have critical values, you can use either a sign diagram or a sketch of the quadratic parabola to find which regions satisfy the inequality. For a positive leading coefficient, the parabola opens upwards, so it is below the x-axis between the roots, and above the x-axis outside the roots.

Methods compared

Two valid exam-accepted methods for finding solution regions:

Sign Diagram

Mark critical values on a number line, test a value in each interval to see if the quadratic is positive or negative, then select intervals matching the inequality sign.

+ Pros: Fast for simple quadratics; No graph drawing required

− Cons: Requires manual testing of values; Easy to mix up signs if you skip steps

Parabola Sketch

Sketch the U-shaped parabola (positive leading coefficient) crossing the x-axis at critical values, then read off regions where the parabola is above or below the x-axis as required.

+ Pros: Visual, lower risk of sign error; Works for all quadratic inequality types

− Cons: Requires basic sketching skills; Takes slightly longer

📐 Worked Example

Solve $x^{2} + x - 6 < 0$ using the sign diagram method.

1
Critical values are $x = - 3$ and $x = 2$ , splitting the number line into 3 intervals: $x < - 3$ , $- 3 < x < 2$ , $x > 2$
2
Test $x = - 4$ : $(- 4 + 3) (- 4 - 2) = 6 > 0$ (does not match < 0)
3
Test $x = 0$ : $(0 + 3) (0 - 2) = - 6 < 0$ (matches inequality)
4
Test $x = 3$ : $(3 + 3) (3 - 2) = 6 > 0$ (does not match < 0)
5
Final solution: $- 3 < x < 2$

Exam tip:

Always check if the inequality is strict ( $<, >$ ) or includes equality ( $\leq, \geq$ ) to know if you should include critical values in your solution set.

4. Common Pitfalls

Wrong move:

Forgetting to flip the inequality sign when dividing/multiplying by a negative number

Why:

Inequalities reverse when multiplied by a negative number because the order of values on the number line flips (e.g. $3 < 5$ becomes $- 3 > - 5$ )

Correct move:

Immediately reverse the inequality direction any time you multiply or divide both sides by a negative value, and double-check this step before submitting your answer.

Wrong move:

Picking the wrong region for quadratic inequalities (e.g. outside roots for $< 0$ with positive leading coefficient)

Why:

Confusion over parabola shape: positive leading coefficient gives a U-shape, so it is below the x-axis between roots, not outside

Correct move:

Memorise that for $a > 0$ : $<$ gives between roots, $>$ gives outside, or draw a quick parabola sketch to confirm the region.

Wrong move:

Forgetting to rearrange all terms to one side before finding critical values for quadratic inequalities

Why:

If terms are on both sides, you cannot correctly factorise to find the roots of the quadratic expression, leading to incorrect critical values

Correct move:

Always rearrange the inequality to standard form $a x^{2} + b x + c ⋈ 0$ with $a > 0$ before factorising or finding critical values.

Wrong move:

Including critical values for strict inequalities ( $<, >$ )

Why:

Strict inequalities mean the expression cannot equal zero, so the roots (where the expression equals zero) are not part of the solution set

Correct move:

Use $<$ or $>$ for strict inequalities (open intervals) and $\leq$ or $\geq$ when equality is allowed (closed intervals).

5. Quick Reference Cheatsheet

Inequality Type	Standard Form ( $a > 0$ )	Solution Region
Linear	$a x + b > c x + d$	Rearrange, flip sign if multiplying/dividing by negative
Quadratic $< / \leq$	$a x^{2} + b x + c < 0$	Single interval: $x_{1} < x < x_{2}$ (or $\leq$ )
Quadratic $> / \geq$	$a x^{2} + b x + c > 0$	Two intervals: $x < x_{1}$ or $x > x_{2}$ (or $\leq / \geq$ )

6. Frequently Asked

Do I flip the inequality sign when multiplying or dividing by a negative number?

Yes! Always reverse the direction of the inequality sign (< ↔ >, ≤ ↔ ≥) if you multiply or divide both sides of an inequality by a negative value. You do not need to flip it for positive values or addition/subtraction of any number.

How do I know if my quadratic inequality solution is one or two intervals?

For a positive leading coefficient: * < or ≤ gives a single interval between the two critical values; * > or ≥ gives two separate intervals on either side of the critical values. This reverses for a negative leading coefficient, or you can multiply the entire inequality by -1 (remember to flip the sign!) to make the leading coefficient positive first.

Going deeper

official_syllabusEdexcel 4PM1 Specification

What's Next

Now that you can solve simple linear and quadratic inequalities, you are ready to move on to more advanced inequality topics in Edexcel IGCSE Further Pure Math. Next, you will learn to graph linear inequalities in two variables for region and linear programming problems, and apply inequality solving to intersection questions in coordinate geometry and roots of quadratics problems. Mastering this foundational topic will also support your work on function domains and optimisation problems in later units. Make sure to practice past paper questions to reinforce your understanding and avoid common mistakes under exam conditions.