Linear and Quadratic Inequalities

Master solving inequalities and expressing solutions in set notation

SPEC 2.5 • ALGEBRA AND FUNCTIONS

What are Inequalities?

An inequality is a mathematical statement that compares two expressions using inequality symbols, showing that one side is greater than, less than, or not equal to the other.

Inequality Symbols

SymbolMeaningExample

<

Less than

x < 5

>

Greater than

x > 3

\leq

Less than or equal to

x \leq 7

\geq

Greater than or equal to

x \geq - 2

Key Difference

Unlike equations (which have specific solutions), inequalities have a range of solutions called a solution set.

Solving Linear Inequalities

Solve linear inequalities using the same techniques as equations, with one critical exception.

You CAN:

Add the same number to both sides

Subtract the same number from both sides

Multiply both sides by a positive number

Divide both sides by a positive number

Critical Rule:

When you multiply or divide by a negative number, you must REVERSE the inequality sign!

Common Mistake

Forgetting to reverse the inequality when dividing by a negative is the most common error. For example:

- 2 x > 6

becomes

x < - 3

(sign reversed!)

Using a Number Line

A number line is a useful visual tool for representing and understanding inequality solutions.

Number Line Conventions

Open circle (○) — endpoint is NOT included (strict inequality: < or >)

Closed circle (●) — endpoint IS included (≤ or ≥)

Arrow — shows the direction of all valid solutions

x > 3

Open circle at 3, arrow pointing right

←——○=========→ 3

x \leq 5

Closed circle at 5, arrow pointing left

←=========●——→ 5

For Combined Inequalities

When solving two inequalities simultaneously, draw both on the same number line and find where they overlap. The overlap region is your solution.

Set Notation

Solutions to inequalities can be written using set notation. This describes the set of all values that satisfy the inequality.

Set Notation Format

{x : condition}

Read as: "the set of all x such that [condition]"

InequalitySet NotationHow to Read It

x > 3

{x : x > 3}

"The set of all x such that x is greater than 3"

x \leq 5

{x : x \leq 5}

"The set of all x such that x is less than or equal to 5"

2 < x < 7

{x : 2 < x < 7}

"The set of all x such that x is between 2 and 7"

x < - 1

or

x > 4

{x : x < - 1} \cup {x : x > 4}

"The union of two sets" (∪ means "or")

Key Symbol

The symbol ∪ (union) is used when combining two separate solution regions, meaning values in either set satisfy the inequality.

Worked Example01

Solve $3 x - 7 > 5$

1

Add 7 to both sides:

3 x - 7 + 7 > 5 + 7

2

Simplify:

3 x > 12

3

Divide both sides by 3 (positive, so sign stays the same):

x > 4

Answer: $x > 4$

On a number line: open circle at 4, arrow pointing right.

Worked Example02

Solve $4 - 2 x \leq 10$ . Give your answer in set notation.

1

Subtract 4 from both sides:

- 2 x \leq 6

2

Divide by −2 (negative, so REVERSE the sign):

x \geq - 3

Answer: ${x : x \geq - 3}$

Dividing by −2 reverses the inequality. On a number line: closed circle at −3, arrow pointing right.

Worked Example03

Find the range of values of x that satisfy both $2 x + 3 > 7$ and $5 x - 1 < 19$

1

Solve the first inequality:

2 x + 3 > 7

2 x > 4

x > 2

2

Solve the second inequality:

5 x - 1 < 19

5 x < 20

x < 4

3

Draw both on a number line:

x > 2

(open at 2, right) and

x < 4

(open at 4, left)

The overlap is where both are satisfied:

2 < x < 4

Answer: $2 < x < 4$

Solving Quadratic Inequalities

Quadratic inequalities require a different approach. We use the shape of the parabola to find solutions.

Method

Step 1: Rearrange to get 0 on one side (standard form)

Step 2: Factorise or find the roots

Step 3: Sketch the parabola (noting where it crosses the x-axis)

Step 4: Read off the solution from the graph

Key Insight

For a positive quadratic

y = (x - a) (x - b)

with roots

a < b

:

y < 0

(below x-axis): solution is

a < x < b

— between the roots

y > 0

(above x-axis): solution is

x < a

or

x > b

— outside the roots

Worked Example04

Solve $x^{2} - 5 x + 6 < 0$

1

Factorise:

x^{2} - 5 x + 6 = (x - 2) (x - 3)

2

Find roots:

x = 2

and

x = 3

3

Sketch: Positive parabola crossing x-axis at 2 and 3

4

We need

< 0

(below x-axis), which is between the roots

Answer: $2 < x < 3$

Worked Example05

Solve $x^{2} - 4 x - 5 \geq 0$ . Give your answer in set notation.

1

Factorise:

x^{2} - 4 x - 5 = (x - 5) (x + 1)

2

Find roots:

x = 5

and

x = - 1

3

Sketch: Positive parabola crossing x-axis at −1 and 5

4

We need

\geq 0

(on or above x-axis), which is outside the roots (including the roots themselves)

Answer: ${x : x \leq - 1} \cup {x : x \geq 5}$

Worked Example06

Solve $2 x^{2} + 3 x - 9 > 0$

1

Factorise:

2 x^{2} + 3 x - 9 = (2 x - 3) (x + 3)

2

Find roots:

2 x - 3 = 0 \Rightarrow x = \frac{3}{2}

and

x + 3 = 0 \Rightarrow x = - 3

3

Sketch: Positive parabola (coefficient of

x^{2}

is positive) crossing at

x = - 3

and

x = \frac{3}{2}

4

We need

> 0

(above x-axis), which is outside the roots

Answer: $x < - 3$ or $x > \frac{3}{2}$

Worked Example07

Find the range of values of x that satisfy both $x^{2} - 9 < 0$ and $x^{2} - 4 x + 3 > 0$ . Give your answer in set notation.

1

Solve

x^{2} - 9 < 0

:

Factorise:

(x - 3) (x + 3) < 0

Roots:

x = - 3

and

x = 3

Below x-axis (between roots):

- 3 < x < 3

2

Solve

x^{2} - 4 x + 3 > 0

:

Factorise:

(x - 1) (x - 3) > 0

Roots:

x = 1

and

x = 3

Above x-axis (outside roots):

x < 1

or

x > 3

3

Combine: Need

- 3 < x < 3

AND (

x < 1

or

x > 3

)

The overlap is

- 3 < x < 1

Answer: ${x : - 3 < x < 1}$

Rational Inequalities

Inequalities involving fractions require careful handling. Never cross-multiply unless you know the sign of both sides!

Method: Critical Values

Step 1: Find critical values (where numerator = 0 and where denominator = 0)

Step 2: Mark these on a number line

Step 3: Test a value in each region to determine the sign

Step 4: Select regions that satisfy the inequality

Critical Warning

Values where the denominator equals zero are never included in the solution (they make the expression undefined).

Worked Example08

Solve $\frac{x - 1}{x + 2} > 0$

1

Find critical values: numerator = 0 when

x = 1

; denominator = 0 when

x = - 2

2

This creates three regions:

x < - 2

,

- 2 < x < 1

,

x > 1

3

Test each region:

x = - 3

:

\frac{- 4}{- 1} = 4 > 0

✓

x = 0

:

\frac{- 1}{2} = - 0.5 < 0

✗

x = 2

:

\frac{1}{4} = 0.25 > 0

✓

4

Solution: regions where expression is positive (excluding

x = - 2

where undefined)

Answer: $x < - 2$ or $x > 1$

Practice

Question 1: Solve

5 - 3 x < 2 x + 15

1

Subtract 2x from both sides:

5 - 5 x < 15

2

Subtract 5 from both sides:

- 5 x < 10

3

Divide by −5 (REVERSE the sign):

x > - 2

Answer: $x > - 2$

Practice

Question 2: Solve

x^{2} + 2 x - 15 \leq 0

. Give your answer in set notation.

1

Factorise:

x^{2} + 2 x - 15 = (x + 5) (x - 3)

2

Roots:

x = - 5

and

x = 3

3

We need

\leq 0

(on or below x-axis), which is between the roots (inclusive)

Answer: ${x : - 5 \leq x \leq 3}$

Practice

Question 3: Find the range of values of x that satisfy both

x + 3 > 1

and

x^{2} - 16 < 0

1

Solve

x + 3 > 1

:

x > - 2

2

Solve

x^{2} - 16 < 0

:

Factorise:

(x - 4) (x + 4) < 0

Between roots:

- 4 < x < 4

3

Draw on number line:

x > - 2

overlaps with

- 4 < x < 4

The overlap is

- 2 < x < 4

Answer: $- 2 < x < 4$

Summary

Key Concepts

TypeMethod

Linear InequalitiesSolve like equations; reverse sign when multiplying/dividing by negative

Quadratic

< 0

Solution is BETWEEN the roots

Quadratic

> 0

Solution is OUTSIDE the roots

Two InequalitiesSolve each separately, use number line to find overlap

RationalFind critical values, test regions, exclude undefined points

Key Exam Skills

Always reverse the inequality sign when multiplying or dividing by a negative number

Use a number line to visualise solutions (open/closed circles, arrows)

Remember:

< 0

means between roots;

> 0

means outside roots

Set notation:

{x : condition}

with ∪ for "or"

For combined inequalities, find where both conditions are satisfied (overlap)