TMUA Inequalities and Modulus

Quadratic inequalities

Factorise, find the roots, and sketch the parabola. For an upward parabola, the expression is negative between the roots and positive outside them.

Linear inequalities

Treat them like equations, except that multiplying or dividing by a negative number reverses the inequality sign.

Modulus

• $|u|<a$ means $-a<u<a$.
• $|u|>a$ means $u<-a$ or $u>a$.
• For $|u|\ge|v|$, squaring both sides is safe because both sides are non-negative.

Using a sketch and test values

A sketch is the fastest check. For $|x-2|<3$, picture the V-shape centred at 2: it lies below 3 between $-1$ and $5$.

For a combined condition such as $x^2>4$ and $x<3$, solve each part separately and take the overlap, which here is $x<-2$ or $2<x<3$. Then test one value from each region. A solution set that fails a test value is wrong, whatever the algebra said.

Squaring an inequality is only safe when both sides are non-negative: $-3<2$ but $9>4$.

For a quadratic inequality with a negative leading coefficient, such as $-x^2+4x-3>0$, multiply by $-1$ and reverse the sign first, or sketch the downward parabola, which is positive between its roots.

Worked examples

Common mistakes

• Not reversing the inequality when multiplying or dividing by a negative.
• Writing $x^2>4$ as $x>2$ and forgetting $x<-2$.
• Multiplying through by an expression whose sign is unknown.
• Giving a one-piece answer to $|u|>a$, which has two separate pieces.

Test yourself on this topic with our free practice questions.