A22AlgebraHigherGrades 7–8

Quadratic Inequalities

Q: What if the x^2 coefficient is negative?

Multiply through by -1 and reverse the inequality, or sketch an upside-down parabola.

Q: Can I use a table of signs instead of a sketch?

Yes, but a sketch is faster and usually clearer.

Q: Do I need set notation?

Only if the question asks for it; inequality notation is otherwise fine.

Solving $x^{2} + 2x - 15 > 0$ is not like solving a linear inequality — the answer is usually two separate regions, or one region trapped between two values. A quick sketch of the parabola shows which, and it removes almost all of the guesswork.

40 min AQA GCSE Maths

The big picture

A quadratic expression is positive where its graph is above the $x$ -axis and negative where it is below. The critical values — the roots — are the points where the sign can change.

For a positive $x^{2}$ coefficient the graph is U-shaped, so it is below the axis between the roots and above the axis outside them. That single picture answers every GCSE quadratic inequality.

What you'll learn

Find the critical values of a quadratic inequality
Use a sketch to decide the solution set
Write solutions using inequality and set notation
Find integer solutions and solve problems in context

The method

Rearrange so one side is 0. Solve the equation to find the critical values.

Sketch a U-shaped curve crossing the $x$ -axis at the critical values.

" $> 0$ " means above the axis (outside the roots); " $< 0$ " means below (between the roots).

Example 1Solve

x^{2} + 2x - 15 > 0

(x + 5)(x - 3) = 0

gives critical values

-5

and

3

2The U-shaped curve is above the axis to the left of

-5

and to the right of

3

Answer

x < -5

x > 3

Your turnSame method, new numbers — try it before moving on.

Solve $x^2 - x - 12 < 0$ .

Your answer

Type it like -2 < x < 5.

Show worked solution

Solution

1 $(x - 4)(x + 3) = 0$ gives critical values $-3$ and $4$ .
2The U-shaped graph is below the axis between them: $-3 < x < 4$ .

Answer: $-3 < x < 4$

Example 2Solve

x^{2} \le 4x

1Rearrange:

x^{2} - 4x \le 0

x(x - 4) = 0

gives

0

and

4

3Below or on the axis between the roots.

Answer

0 \le x \le 4

Your turnSame method, new numbers — try it before moving on.

Solve $x^2 \ge 9$ .

Your answer

Type it like x <= -3 or x >= 3.

Show hint

Rearrange to $x^2 - 9 \ge 0$ . Do not just write $x \ge 3$ .

Show worked solution

Solution

1 $x^2 - 9 \ge 0$ : critical values $\pm 3$ .
2The graph is above the axis outside the roots: $x \le -3$ or $x \ge 3$ .

Answer: $x \le -3$ or $x \ge 3$

Tip — Never divide both sides by $x$ — it may be negative, and you lose the solution $x = 0$ .

Writing the answer

Between the roots is a single inequality: $a < x < b$ .

Outside the roots needs two parts: $x < a$ or $x > b$ . It cannot be written as one inequality.

In set notation: $\{x : x < -5\} \cup \{x : x > 3\}$ .

Why this works

Writing $3 < x < -5$ is impossible — no number is both greater than 3 and less than −5. If your answer reads like that, the solution must be two separate regions.

Integer solutions and problems

Once the solution set is found, list the integers it contains.

Context questions often give an area or cost that must exceed a value, producing a quadratic inequality.

Example 1Find the integers that satisfy

x^{2} - 7x + 10 \le -2

x^{2} - 7x + 12 \le 0

(x - 3)(x - 4) = 0

gives 3 and 4.

3Solution:

3 \le x \le 4

Answer

x = 3

and

x = 4

Your turnSame method, new numbers — try it before moving on.

List the integers that satisfy $x^2 - 5x < 0$ .

Your answer

Separate values with commas.

Show worked solution

Solution

1 $x(x - 5) < 0$ gives $0 < x < 5$ .
2Integers: $1, 2, 3, 4$ .

Answer: $1, 2, 3, 4$

Think like an examiner

Show the critical values clearly.
Include a sketch — it often earns a method mark.
Use strict or inclusive inequalities to match the question.

Remember these

(x - a)(x - b) < 0 \Rightarrow a < x < b

Below the axis: between the roots.

(x - a)(x - b) > 0 \Rightarrow x < a \text{ or } x > b

Above the axis: outside the roots.

b^{2} - 4ac

Discriminant: decides the number of roots.

Watch out for these

Answering

x > -5

and

x > 3

from the factors.

Use a sketch to decide the regions.

Writing

3 < x < -5

x < -5

x > 3

Dividing by

x

x^{2} \le 4x

Rearrange to

x(x - 4) \le 0

Changing

\le

<

in the answer.

Keep the inequality type from the question.

Stretch yourself

The equation $x^{2} + kx + 9 = 0$ has no real solutions. Find the range of values of $k$ .

Hint — No real solutions means the discriminant is negative.

Reveal solution

SolutionNo real roots requires

b^{2} - 4ac < 0

k^{2} - 36 < 0

. Critical values:

k = -6

and

k = 6

. The graph of

k^{2} - 36

is below the axis between them, so

-6 < k < 6

\blacksquare

Questions students ask

What if the

x^{2}

coefficient is negative?

Multiply through by

-1

and reverse the inequality, or sketch an upside-down parabola.

Can I use a table of signs instead of a sketch?

Yes, but a sketch is faster and usually clearer.

Do I need set notation?

Only if the question asks for it; inequality notation is otherwise fine.

Key takeaways

Rearrange to compare with 0 and find the critical values.
Sketch the parabola through the roots.
$< 0$ : between the roots. $> 0$ : outside the roots.
Outside regions need two inequalities joined by "or".

Exam-style questions

Practise it the way it is examined

2 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 73 marks

Solve $2x^2 - 5x - 3 \le 0$ .[3]
Show mark scheme
- M1Factorises: $(2x + 1)(x - 3)$
- M1Critical values $-\frac{1}{2}$ and $3$
- A1 $-\frac{1}{2} \le x \le 3$
Answer: $-\frac{1}{2} \le x \le 3$

Question 2Grade 83 marks

Find the values of $x$ that satisfy both $3x - 1 > 5$ and $x^2 - 6x + 8 < 0$ .[3]
Show mark scheme
- B1 $x > 2$
- M1 $(x - 2)(x - 4) < 0$ gives $2 < x < 4$
- A1Combined: $2 < x < 4$
Answer: $2 < x < 4$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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