Inequalities

The idea

Solve a linear inequality like an equation, flipping the sign if you multiply or divide by a negative. For a quadratic inequality, find the roots and sketch: the curve is below the axis between the roots and above it outside them.

Rearrange so one side is $0$ and find the critical values.
Sketch the graph to see where it is above or below the $x$ -axis.
Write the set: one interval such as $1 \le x \le 4$ , or two parts such as $x < -2$ or $x > 4$ .

Watch out Give a range of values, not just the critical values.

See it worked

Worked example

Solve $x^2 - 2x - 8 > 0$ .

$(x - 4)(x + 2) = 0$ gives critical values $-2$ and $4$ .
The U-shaped curve is above the axis outside the roots.
$x < -2$ or $x > 4$ .

Answer $x < -2$ or $x > 4$

Your turn

Question 1Non-calculator3 marks

Solve $3x - 7 > 5 - x$ .

Question 2Non-calculator3 marks

Solve $x^2 - 5x + 4 \le 0$ .

Question 3Non-calculator3 marks

Solve $2x^2 + x - 6 < 0$ .

In past papers

47 official questions in 41 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

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