CDS Mathematics · Inequalities

Solving Inequalities

Solve an inequality like an equation, but reverse the sign when multiplying or dividing by a negative; a factored quadratic is positive outside its roots and negative between them.

All Inequalities notes CDS Mathematics strategy

Why this matters

Five PYQs, none HARD. Linear pairs: solve each and keep the overlap. Quadratics: factorise, mark the roots on a line, and read the sign of the product in each stretch. One item asks which quadrants a region reaches — test a point in each.

Concept 1 of 1: Linear and quadratic inequalities

A product of two factors is positive when they have the same sign. With roots

\alpha < \beta

(x - \alpha)(x - \beta)

is positive to the left of

\alpha

and to the right of

\beta

, and negative in between.

Definition

Multiplying or dividing by a negative number reverses the inequality.
Two conditions at once: solve each, keep the values that satisfy both.
$(x - \alpha)(x - \beta) > 0$ : $x < \alpha$ or $x > \beta$ . $< 0$ : $\alpha < x < \beta$ .
A region in the plane: test one point in each quadrant.

Quadratic sign

(x - \alpha)(x - \beta) > 0 \iff x < \alpha \text{ or } x > \beta

Worked example

Solve

3x - 4 < 2x + 1

and

4x + 3 > x + 9

together.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q59Easy

Example 1 · Inequalities · Solving Linear and Quadratic Inequalities

What is the solution of the inequalities

5x + 3 < 8x - 9

and

2x + 20 > 5x + 2

'Or', not 'and', outside the roots

x^2 - 6x - 27 > 0

holds for

x > 9

x < -3

. Writing '

x < 9

x > -3

' covers every number and is the distractor.

Drill 4 more on linear and quadratic inequalities

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Linear and quadratic inequalities
Quadratic sign
$(x - \alpha)(x - \beta) > 0 \iff x < \alpha \text{ or } x > \beta$

Watch out for (1)

'Or', not 'and', outside the roots
→ Linear and quadratic inequalities

Test yourself on a real paper

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