Quadratic Inequalities

A quadratic inequality has a range of solutions, not just two values. You find the critical values, decide which regions work, and write the answer in the notation the question asks for.

Critical values

Move everything to one side, with 0 on the other.
Factorise.
The critical values are the roots. They split the number line into regions.

Example. $x^{2} + 2 x \geq 15 \Rightarrow x^{2} + 2 x - 15 \geq 0 \Rightarrow (x + 5) (x - 3) \geq 0$

Critical values: $x = - 5$ and $x = 3$ .

Never divide by an expression containing $x$ , because its sign may be negative. In $(2 x - 1) (x + 3) > (x + 3)$ , move $(x + 3)$ across and factorise instead.

Number line and sketch methods

Sketch method. Sketch the parabola through its roots.

$a > 0$ : negative between the roots, positive outside.
$a < 0$ : positive between the roots, negative outside.

Number line method. Test a value in each region in the factorised form.

Example. $(x + 5) (x - 3) \geq 0$ with $a > 0$ : positive outside the roots, so $x \leq - 5$ or $x \geq 3$ .

Example. $- x^{2} + x + 6 > 0$ . Multiply by $- 1$ and reverse the sign: $x^{2} - x - 6 < 0$ , so $(x - 3) (x + 2) < 0$ , giving $- 2 < x < 3$ .

Interval and set notation

Inequality notation: $- 2 < x < 3$
Interval notation: $x \in (- 2; 3)$ . Round brackets exclude endpoints, square brackets include them.
Set-builder notation: ${x : - 2 < x < 3, x \in R}$

"Or" answers: $x \leq - 5$ or $x \geq 3$ is $x \in (- \infty; - 5] \cup [3; \infty)$ .

Infinity always takes a round bracket.

Integer solutions. If $- 2 < x < 3$ and $x$ must be an integer, the solutions are $- 1, 0, 1, 2$ .

Key terms

Critical values: The roots that separate the regions of the number line.
Interval notation: Writing a range of values with brackets, such as $(- 2; 3]$ .
Union: Combining two intervals, written $\cup$ .

Common mistakes

Dividing by an expression with $x$ in it.
Forgetting to reverse the sign when multiplying by a negative.
Using square brackets with infinity.
Writing 'or' solutions as one combined inequality.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

3 marks total

knowledge

Solve $x^{2} - 9 < 0$ and write the answer in interval notation.

Worked solution

Hidden - try the question first

Solution

$(x - 3) (x + 3) < 0$
$- 3 < x < 3$
$x \in (- 3; 3)$

Question 2

5 marks total

routine

Solve for $x$ : $x^{2} + 2 x \geq 15$

Worked solution

Hidden - try the question first

Solution

$(x - 3) (x + 3) < 0$
$(x + 5) (x - 3) \geq 0$
Critical values $- 5$ and $3$
$x \leq - 5$
or $x \geq 3$

Question 3

6 marks total

complex

(a) Solve $- x^{2} + x + 6 > 0$ . (4)

(b) List the integers that satisfy the inequality. (2)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - x - 6 < 0$ (multiply by $- 1$ , reverse the sign)
$(x - 3) (x + 2) < 0$
$- 2 < x$
$< 3$
(b) $- 1, 0$
$1, 2$

Question 4

8 marks total

problem-solving

(a) Solve $x (x - 4) \leq - 3$ . (3)

(b) Solve $(2 x - 1) (x + 3) > x + 3$ . (3)

Worked solution

Hidden - try the question first

Solution

(a) $x^{2} - x - 6 < 0$
$(x - 1) (x - 3) \leq 0$
$- 5$
(b) $(2 x - 1) (x + 3) - (x + 3) > 0$
$(x + 3) (2 x - 2) > 0$
$x < - 3$ or $x > 1$
(c) $x \in (- \infty; - 3)$
$\cup (1; \infty)$

FAQs

How do I know whether the solution is between or outside the roots?

Sketch the parabola. For $a > 0$ it is below the axis between the roots and above it outside.

Why can't I divide by (x + 3)?

If $x + 3$ is negative, dividing would reverse the inequality, and you would not know which way to write it. Factorise instead.

Which notation should I use?

Use the notation the question asks for. If none is given, inequality notation is fine.

Critical values

Number line and sketch methods

Interval and set notation

Key terms

Common mistakes

Practice questions

FAQs

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