Intermediate Algebra/Absolute Value Inequalities

Compound Inequalities

In the previous section we dealt with inequalities with one specific constraint - that is, only one solution set. However, there is a good chance that real-world situations will require multiple constraints. For example, computer manufacturers will want their products to be sold within certain ranges so that customers will continue to buy their computers.

Such situations are also bound to happen in algebra, and can be represented using compound inequalities. A compound inequality is a statement that puts two constraints on a constant, variable, or expression. For example, the inequality -4 < x < 3 indicates that x has two constraints: it must be (a) greater than -4 and (b) less than 3. These compound inequalities are often referred to as "and" inequalities because they require that the expression in the center meets the two restraints.

Using the graphing methods covered in the last lesson, we can show the solution set of -4 < x < 3 on the number line:

                                    O--------------------O
                              <--|--|--|--|--|--|--|--|--|--|--|-->
                                -5 -4 -3 -2 -1  0  1  2  3  4  5

Notice that the solution is bounded by the two values.

On the other hand, a compound inequality can have one out of two constraints. These inequalities are called "or" inequalities - the expression can satisfy either one of the conditions. If, for example, x < 3 or x $\geq$ 7, then x is equal to any number less than 3 or greater than or equal to 7. Any number between 3 and 7 will not satisfy either condition.

Represented graphically, the inequality x < 3 or x $\geq$ 7 looks like this:

                     <--------------------------------O           *-->
                     <--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->
                       -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7

Notice that, as opposed to the inequality in the previous example, the two constraints travel in opposite directions. This is a key principle of compound inequalities, and one that should be kept in mind when graphing solutions.

Solving Compound Inequalities

Because compound inequalities are two separate constraints combined into one, all operations must be performed on all sides of the inequality.

Example 1: Solve for x: -7 < 2x + 4 $\leq$ 12.

Stop for a second and pretend that the statement above was two separate inequalities. The logical first step would be to subtract 4 from both sides. Because the two are now put into one, however, we must subtract 4 from all sides. Therefore:

-7 < 2x + 4 $\leq$ 12

-7 - 4 < 2x + 4 - 4 $\leq$ 12 - 4

-11 < 2x $\leq$ 8

Now divide all sides by 2.

-11 < 2x $\leq$ 8

${\frac {-11}{2}}<{\frac {2x}{2}}\leq {\frac {8}{2}}$

-5.5 < x $\leq$ 4

Example 2: Solve for x: -3 > -x + 7 or 2x - 6 $\leq$ -4.

Because the two inequalities act as separate entities, solve each one separately. Start by subtracting 7 from both sides in the first inequality, and add 6 to both sides in the second.

               -3 > -x + 7                  or                    2x - 6 $\leq$ -4
    
              -10 > -x                                                2x $\leq$ 2

Finish by dividing by respective coefficients (-1 and 2).

              -10 > -x                                              2x $\leq$ 2

                x > 10                      or                       x $\leq$ 1