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How to Find the Interval of an Inequality

When you find the interval of an inequality, you might be asked to show the answer on a number line. If there are more than one inequality, you’ll have to show where all of them are true. This is how you do it:

Rule

Finding the Intervals of Inequalities

1.
Solve the inequality/inequalities.
2.
If you have only one inequality, draw a sign chart and show the interval where the inequality is true with the help of an arrow.
3.
If you have more than one inequality, draw sign charts for all of them underneath each other.
4.
Mark the intervals where all the inequalities are true.

Example 1

Solve the inequality 3x 3 > 15 + x and draw the interval

3x 3 > 15 + x 3x x > 15 + 3 2x > 18 |÷ 2 x > 9 Draw the sign charts for the solution and read the interval off the sign chart.

A sign chart where 9 is marked. The interval from 9 to infinity is marked as the solution.

You can see that the inequality is true in the interval (9,).

Example 2

Find the intervals where the expressions 2x + 3 < 1 and 2x 3 < 1 are both true

Begin by setting up and solving the inequalities: 2x + 3 < 1 2x 3 < 1 2x < 2|÷ 2 2x < 4|÷ (2) x < 1 x > 2

Then you can draw the sign charts for the inequalities and read off where they’re both true.

The solutions x<-1 and x>-2 are drawn in a sign chart and used to find the solution to the system of inequalities.

You can see that both of the inequalities are true on the interval (2,1).

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How Do Sign Charts Work?