Inequalities

Quadratic Inequalities

A quadratic inequality may look like:

or

The main difference from a quadratic equation is that we are looking for a range of values.

Method

Move everything to one side.

Factorise.

Find the critical values/roots.

Divide the number line into intervals.

Test the intervals or use the sign pattern.

Write the solution.

Represent the solution on a number line.

Worked Example 1

Solve:

Step 1: Factorise

Therefore:

Step 2: Find the critical values

These divide the number line into:

Step 3: Determine the signs

Interval

�

�

Product

�

−

−

+

�

+

−

−

�

+

+

+

We need the product to be positive.

Therefore:

Because the inequality is �, 2 and 3 are not included.

Example 2

Solve:

Factorise:

Critical values:

The expression is negative between the roots.

Because the inequality includes equality:

Important rule

For:

� or � → use open circles

� or � → use closed circles

8. INEQUALITIES IN TWO DIMENSIONS

A two-dimensional inequality contains both � and �.

Examples:

Steps for solving graphically

Step 1: Draw the boundary line

For:

first draw:

Step 2: Decide whether the line is solid or broken

� or � → broken/dashed line

� or � → solid line

Step 3: Choose a test point

Usually use:

Substitute:

This is false.

Therefore, the side containing � is not shaded.

Step 4: Shade the correct half-plane

The shaded region contains all points satisfying the inequality.