Quadratic inequalities by testing values Revision Card

AI generatedAlgebraGCSE Higher / IGCSE Extended

Quadratic inequalities by testing values

Rearrange so that one side is zero before finding the roots.
The roots divide the number line into intervals. Choose intervals where the quadratic has the required sign.
Use open endpoints for strict inequalities and closed endpoints when equality is included. A solution is a set of values, not just the roots.

Example 1

Solve the inequality by testing a value in each interval.

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the positive intervals. The sign cannot change between consecutive roots. Exclude both roots because the inequality is strict.

Example 2

Solve the inequality by testing a value in each interval.

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the negative intervals. The sign cannot change between consecutive roots. Include both roots because equality is allowed.

Your turn

Question 1

Solve the inequality by testing a value in each interval.

Check answer

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the negative intervals. The sign cannot change between consecutive roots. Exclude both roots because the inequality is strict.

Question 2

Solve the inequality by testing a value in each interval.

Check answer

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the positive intervals. The sign cannot change between consecutive roots. Include both roots because equality is allowed.

Question 3

Solve the inequality by testing a value in each interval.

Check answer

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is negative.

Test

in the middle interval. The result is positive.

Test

in the right interval. The result is negative. Choose the positive intervals. The sign cannot change between consecutive roots. Exclude both roots because the inequality is strict.

Question 4

Solve the inequality by testing a value in each interval.

Check answer

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the negative intervals. The sign cannot change between consecutive roots. Include both roots because equality is allowed.

Question 5

Solve the inequality by testing a value in each interval.

Check answer

Find the boundary values by solving the corresponding equation with zero.

Factorise to find where the graph meets the

-axis.

Set each factor equal to zero:

. These values divide the number line into three intervals.

Test

in the left interval. The result is positive.

Test

in the middle interval. The result is negative.

Test

in the right interval. The result is positive. Choose the negative intervals. The sign cannot change between consecutive roots. Exclude both roots because the inequality is strict.