Do you know how to graph an inequality with more than one inequality symbol, like \begin{align*}\text{-}4 < x < 5?\end{align*}

### Compound Inequalities

Compound inequalities are 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. These compound inequalities may appear in the form \begin{align*}x>a \text{ and } x <b,\end{align*} but are commonly are combined, like \begin{align*}a<x<b.\end{align*} Possible values for \begin{align*}x\end{align*} 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 “\begin{align*}x>a\end{align*} or \begin{align*}x<b\end{align*}.” Possible values for \begin{align*}x\end{align*} are ones that will make*at least one*of the inequalities true.

You might wonder why the variable \begin{align*}x\end{align*} has to be *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 “\begin{align*}x>5\end{align*} and \begin{align*}x>3\end{align*}.” Are there any numbers greater than 5 that are *not* greater than 3? No! Since 5 is greater than 3, everything greater than 5 is also greater than 3. If we say \begin{align*}x\end{align*} is greater than both 5 and 3, that doesn’t tell us any more than if we just said \begin{align*}x\end{align*} is greater than 5. So this compound inequality isn’t really compound; it’s equivalent to the simple inequality \begin{align*}x > 5\end{align*}. And that’s what would happen no matter which two numbers we used; saying that \begin{align*}x\end{align*} is greater than both numbers is just the same as saying that \begin{align*}x\end{align*} is greater than the bigger number, and saying that \begin{align*}x\end{align*} is less than both numbers is just the same as saying that \begin{align*}x\end{align*} is less than the smaller number.

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 “\begin{align*}x>5\end{align*} or \begin{align*}x>3\end{align*},” that’s the same as saying just “\begin{align*}x>3\end{align*}.” Saying that \begin{align*}x\end{align*} is greater than at least one of two numbers is just the same as saying that \begin{align*}x\end{align*} is greater than the smaller number, and saying that \begin{align*}x\end{align*} is less than at least one of two numbers is just the same as saying that \begin{align*}x\end{align*} is less than the greater number.

**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 \begin{align*}x \ge -40\end{align*} *and* \begin{align*}x<60\end{align*}.

This is usually written as \begin{align*}-40 \le x < 60\end{align*}.

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: \begin{align*}x>1\end{align*} *or* \begin{align*}x < -2\end{align*}.

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: \begin{align*}x \ge 4\end{align*} *or* \begin{align*}x \le -1\end{align*}.

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

a) \begin{align*}-4 \le x \le 6\end{align*}

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

b) \begin{align*}x < 0\end{align*} or \begin{align*}x > 2\end{align*}

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

c) \begin{align*}x \ge -8\end{align*} or \begin{align*}x \le -20\end{align*}

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) \begin{align*}5x+2(x-3) \ge 2\end{align*}

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

(To get the \begin{align*}\ge\end{align*} symbol, press **[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 \begin{align*}y-\end{align*}value jumping from 0 to 1.

The solution set is the values of \begin{align*}x\end{align*} for which the graph shows \begin{align*}y=1\end{align*}—in other words, the set of \begin{align*}x-\end{align*}values that make the inequality true.

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 \begin{align*}x>\frac{8}{7}\end{align*}, which is why you can see the \begin{align*}y-\end{align*}value changing from 0 to 1 at about 1.14.

b) \begin{align*}7x-2 < 10x+1 < 9x+5\end{align*}

This is a compound inequality: \begin{align*}7x-2 < 10x +1\end{align*} and \begin{align*}10x+1 < 9x+5\end{align*}. You enter it like this:

(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 \begin{align*}x\end{align*} for which \begin{align*}y=1\end{align*}; in this case that would be \begin{align*}-1<x<4\end{align*}.

c) \begin{align*}3x+2 \le 10\end{align*} or \begin{align*}3x+2 \ge 15\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*} for which \begin{align*}y=1\end{align*}--in this case, \begin{align*}x \le 2.7\end{align*} or \begin{align*}x \ge 4.3\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*}
- \begin{align*}x < 7\end{align*} or \begin{align*}x > 25\end{align*}
- \begin{align*}x \ge -100\end{align*} or \begin{align*}x \le -50\end{align*}
- \begin{align*}-1 < x < 200\end{align*}
- \begin{align*}2000 < x \le 2001\end{align*}
- \begin{align*}x \le 1.56\end{align*} or \begin{align*}x > 1.78\end{align*}
- \begin{align*}x > 0.0005\end{align*} or \begin{align*}x \le -0.03\end{align*}

### Review (Answers)

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