# Algebra: Compound Inequalities

## Compound Inequalities

You can define a range of values using a *single* inequality statement, called a *compound inequality*. Rather than say that, for a given equation, *x* > 2 and also *x* ¤ 9, you could write 2 < *x* < 9. This literally reads "2 is less than *x*, which is less than or equal to 9," but you might understand it better if I reword it like this: "*x* has a value between 2 and 9it might even be 9, but it can't be 2."

##### Talk the Talk

A **compound inequality** is a single statement used to represent two inequalities at once, such as *a* < *x* < *b*. It describes a range (or *interval*) of numbers, whose lower boundary is *a*, and whose upper boundary is *b*. Whether or not those numbers are actually included in the interval is decided by the attached inequality sign. If the symbol is ¤, then the number is included; if it's <, then the boundary is not included.

In Figure 7.3, I illustrate the parts of a compound inequality. Notice that you should always write the lower boundary on the left and the upper boundary on the right. Furthermore, you should always use either the < or ¤ symbol, or the statement may not make sense.

### Solving Compound Inequalities

Notice that a compound inequality has three pieces to it, rather than the usual two sides of an equation or simple inequality. In order to solve a compound inequality, your job is to isolate the variable in the middle of the statement. Do this by adding, subtracting, multiplying, and dividing the same thing to *all three parts* of the inequality at the same time.

##### Kelley's Cautions

If you have to divide all three parts of a compound inequality by a negative number, you'll have to reverse *both* inequality signs to end up with this: *b* > *x* > *a*, where *b* is the upper boundary and *a* is the lower boundary. That's okay, but I prefer it rewritten as *a* < *x* < *b*, with less than signs and the lower boundary back on the left, where it belongs.

**Example 4**: Solve the inequality - 4 ¤ 3*x* + 2 < 20.

**Solution**: You want to isolate the *x* where the 3*x* +2 now is, so start by subtracting 2, not only from there, but from all three parts of the inequality.

- -6 ¤ 3
*x*< 18

To eliminate the *x*'s coefficient, divide *everything by 3.*

*-2 ¤**x*< 6

*This means any number between -2 and 6 (including -2 but excluding 6) will make the compound inequality true.*

*You've Got Problems*

*You've Got Problems*

*Problem 4: Solve the inequality -1 < 2 x + 5 < 13.*

*Graphing Compound Inequalities*

*Graphing Compound Inequalities*

*To graph a compound inequality, use dots to mark its endpoints on a number line. Just like in basic inequality graphs, the dots of compound inequalities correspond with the type of inequality symbol. Specifically, the symbol ¤ should be marked with a solid dot, and the symbol < should be marked with an open dot. Since a compound inequality statement has two inequality symbols, use the one closest to each endpoint to help you decide what sort of dot to draw for that endpoint.*

*Once you've drawn the dots, draw a dark segment connecting them. This dark segment indicates that all the numbers between the endpoints are solutions to the inequality.*

**Example 5**: Graph the compound inequality -14 < *x* ¤ -6.

**Solution**: Since -14, the lower endpoint of this interval, has a < symbol next to it, draw an open dot at -14 on the number line. The upper boundary, on the other hand, has a ¤ symbol next to it, so mark -6 with a solid dot. Now, connect the two dots with a dark line, as shown in Figure 7.4.

*You've Got Problems*

*You've Got Problems*

*Problem 5: Graph the compound inequality -2 ¤ x < 5.*

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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