# 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

### FindingtheIntervalsofInequalities

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

$\begin{array}{llllll}\hfill 3x-3& >15+x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 3x-x& >15+3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2x& >18\phantom{\rule{2em}{0ex}}& \hfill & |÷2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& >9\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ Draw the sign charts for the solution and read the interval off the sign chart.

You can see that the inequality is true in the interval $\phantom{\rule{-0.17em}{0ex}}\left(9,\infty \right)$.

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: $\begin{array}{llllllll}\hfill 2x+3& <1\phantom{\rule{2em}{0ex}}& \hfill -2x-3& <1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2x& <-2\phantom{\rule{1em}{0ex}}|÷2\phantom{\rule{2em}{0ex}}& \hfill -2x& <4\phantom{\rule{2em}{0ex}}|÷\left(-2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& <-1\phantom{\rule{2em}{0ex}}& \hfill x& >-2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

You can see that both of the inequalities are true on the interval $\phantom{\rule{-0.17em}{0ex}}\left(-2,-1\right)$.