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 $3 x - 3 > 15 + x$ and draw the interval

\begin{array}{l} 3 x - 3 & > 15 + x \\ 3 x - x & > 15 + 3 \\ 2 x & > 18 & | \div 2 \\ x & > 9 \end{array}

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 $(9, \infty)$ .

Example 2

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

Begin by setting up and solving the inequalities:

\begin{array}{l} 2 x + 3 & < 1 & - 2 x - 3 & < 1 \\ 2 x & < - 2 | \div 2 & - 2 x & < 4 | \div (- 2) \\ x & < - 1 & x & > - 2 \end{array}

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?

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