Do you know how to graph an inequality with more than one inequality symbol, like

### Compound Inequalities

In this section, we’ll solve compound inequalities—inequalities with more than one constraint on the possible values the solution can have.

There are two types of compound inequalities:

- Inequalities joined by the word
*“and,”*where the solution is a set of values greater than a number*and*less than another number. We can write these inequalities in the form “x>a andx<b ,” but usually we just write “a<x<b .” Possible values forx are ones that will make*both*inequalities true. - Inequalities joined by the word
*“or,”*where the solution is a set of values greater than a number*or*less than another number. We write these inequalities in the form “x>a orx<b .” Possible values forx are ones that will make*at least one*of the inequalities true.

You might wonder why the variable *greater than* one number and/or *less than* the other number; why can’t it be greater than both numbers, or less than both numbers? To see why, let’s take an example.

Consider the compound inequality “*not* greater than 3? No! Since 5 is greater than 3, everything greater than 5 is also greater than 3. If we say

Compound inequalities with “or” work much the same way. Every number that’s greater than 3 *or* greater than 5 is also just plain greater than 3, and every number that’s greater than 3 is certainly greater than 3 *or* greater than 5—so if we say “

**Write and Graph Compound Inequalities on a Number Line**

*Write the inequalities represented by the following number line graphs.*

a)

The solution graph shows that the solution is any value between -40 and 60, including -40 but not 60.

Any value in the solution set satisfies both *and*

This is usually written as

b)

The solution graph shows that the solution is any value greater than 1 (not including 1) or any value less than -2 (not including -2). You can see that there can be no values that can satisfy both these conditions at the same time. We write: *or*

c)

The solution graph shows that the solution is any value greater than 4 (including 4) or any value less than -1 (including - 1). We write: *or*

#### Graph the following compound inequalities on a number line.

a)

The solution is all numbers between -4 and 6, including both -4 and 6.

b)

The solution is all numbers less than 0 or greater than 2, not including 0 or 2.

c)

The solution is all numbers greater than or equal to -8 or less than or equal to -20.

**Solve Compound Inequalities Using a Graphing Calculator (TI-83/84 family)**

Graphing calculators can show you the solution to an inequality in the form of a graph. This can be especially useful when dealing with compound inequalities.

a)

Press the **[Y=]** button and enter the inequality on the first line of the screen.

(To get the **[TEST] [2nd] [MATH]** and choose option 4.)

Then press the **[GRAPH]** button.

Because the calculator uses the number 1 to mean “true” and 0 to mean “false,” you will see a step function with the

The solution set is the values of

Note: You may need to press the **[WINDOW]** key or the **[ZOOM]** key to adjust the window to see the full graph.

The solution is

b)

This is a compound inequality:

(To find the **[AND]** symbol, press **[TEST]**, choose **[LOGIC]** on the top row and choose option 1.)

The resulting graph should look like this:

The solution are the values of

c) \begin{align*}3x+2 \le 10\end{align*}

This is another compound inequality.

(To enter the **[OR]** symbol, press **[TEST]**, choose **[LOGIC]** on the top row and choose option 2.)

The resulting graph should look like this:

The solution are the values of \begin{align*}x\end{align*}

### Examples

#### Example 1

*Write the inequality represented by the following number line graph.*

The solution graph shows that the solution is any value that is both less than 25 (not including 25) and greater than -25 (not including -25). Any value in the solution set satisfies both \begin{align*}x >-25\end{align*}*and* \begin{align*}x < 25.\end{align*}

This is usually written as \begin{align*}-25 < x < 25.\end{align*}

#### Example 2

*Graph the following compound inequality on a number line.*

\begin{align*}-15 < x \le 85\end{align*}

The solution is all numbers between -15 and 85, not including -15 but including 85.

### Review

Write the compound inequalities represented by the following graphs.

Graph the following compound inequalities on a number line.

- \begin{align*}-2 \le x \le 20\end{align*}
−2≤x≤20 - \begin{align*}x < 7\end{align*}
x<7 or \begin{align*}x > 25\end{align*}x>25 - \begin{align*}x \ge -100\end{align*}
x≥−100 or \begin{align*}x \le -50\end{align*}x≤−50 - \begin{align*}-1 < x < 200\end{align*}
−1<x<200 - \begin{align*}2000 < x \le 2001\end{align*}
2000<x≤2001 - \begin{align*}x \le 1.56\end{align*}
x≤1.56 or \begin{align*}x > 1.78\end{align*}x>1.78 - \begin{align*}x > 0.0005\end{align*}
x>0.0005 or \begin{align*}x \le -0.03\end{align*}x≤−0.03

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.5.