MathematicsChapter 4: Linear and Quadratic Inequalities

Quadratic Inequalities

For (x-p)(x-q)>0 the product is positive outside the roots (same sign).

Learning objectives

Explain quadratic inequalities using the correct definitions, conditions, and notation.
Apply outside vs between, test a point correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.

Mathematical form

test x = -2: (-)(-) > 0 true

test x = 0: (+)(-) < 0 false

test x = 4: (+)(+) > 0 true

x = -1 and x

Where / symbols

QUADRATIC INEQUALITIES: For (x-p)(x-q)>0 the product is positive outside the roots (same sign). For <0 it is negative between the roots (opposite signs). If the parabola opens up (a>0), >0 is outside and <0 is inside. Sketch a sign chart: test a point in each interval.
PRODUCT FORM: (x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints are included.
FROM STANDARD FORM: x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10. x^2+4x-21<0 is (x+7)(x-3)<0 so -7<x<3.

Overview

Roots first, then signs:

Product Form

(x+1)(x-3)>0 when x<-1 or x>3. (x-9)(x+1)<0 when -1<x<9. Equal roots and = change whether endpoints are included.

Examples x<-1 or x>3 · -1<x<9 · Include roots only for <= or >=

From Standard Form

x^2 - x - 90 > 0 factors as (x-10)(x+9)>0, so x<-9 or x>10. x^2+4x-21<0 is (x+7)(x-3)<0 so -7<x<3.

Examples (x-10)(x+9)>0 · x^2+4x-21<0 · a>0 opens up

WHAT — QUADRATIC INEQUALITIES

WHY — Why this topic matters

Quadratic Inequalities matters because inequalities describe ranges and limits rather than one exact value. They are used for constraints, graphs, optimisation, and real situations such as budgets, dimensions, and capacity.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through PRODUCT FORM, then FROM STANDARD FORM. Worked: solve (x + 1)(x - 3) > 0. roots: x = -1 and x = 3 intervals: x < -1, -1 < x < 3, x > 3 test x = -2: (-)(-) > 0 true test x = 0: (+)(-) < 0 false test x = 4: (+)(+) > 0 true solution: x < -1 or x > 3 Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

solve (x + 1)(x - 3) > 0.
roots: x = -1 and x = 3
intervals: x < -1, -1 < x < 3, x > 3
test x = -2: (-)(-) > 0 true
test x = 0: (+)(-) < 0 false
test x = 4: (+)(+) > 0 true
solution: x < -1 or x > 3

In simple words

Do not solve (x+1)(x-3)>0 as if it were =0 and then 'the roots are the answer'. The roots are the boundaries, not the solution of a strict inequality.

Key points

outside vs between
test a point
>0 outside if a>0
factor then chart

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.