Defining Compound Inequalities

A compound inequality is of the following form:

$a < x < b$

There are actually two statements here. The first statement is $a < x$. The next statement is $x < b$. This statement is therefore read as "$a$ is less than $x$, and $x$ is less than $b$."

The compound inequality $a < x < b$ indicates "betweenness"—the number $x$ is between the numbers $a$ and $b$. Without changing the meaning, the statement $a<x$ can also be read as $x>a$. Therefore, the form $a < x < b$ can also be read as "$x$ is greater than $a$, and at the same time is less than $b$." 

Consider $4 < x < 9$. This states that $x$ is some number strictly between 4 and 9. For a visualization of this inequality, refer to the number line below. The numbers 4 and 9 are not included, so we place open circles on these points.

$4 < x < 9$

The above inequality on the number line.

Similarly, consider $-2 < z < 0$. In this case, $z$ is some number strictly between -2 and 0. Again, because the numbers -2 and 0 are not included, we place open circles on those points. 

$-2 < x < 0$

The above inequality on the number line.

$$Solving Compound Inequalities

Now consider $1 < x + 6 < 8$. The expression $x + 6$ represents some number strictly between 1 and 8. However, the meaning of this is difficult to visualize—what does it mean to say that an expression, rather than a number, lies between two points? Not to worry—we can still find all possible values of not only the expression, but the variable $x$ itself.

The statement $1 < x + 6 < 8$ says that the quantity $x + 6$ is between 1 and 8, a statement that will be true for only certain values of $x$. 

To solve for possible values of $x$, we need to get $x$ by itself:

$1 - 6 < x + 6 - 6 < 8 - 6$

$-5 < x < 2$

Therefore, we find that if $x$ is any number strictly between -5 and 2, the statement $1 < x + 6 < 8$ will be true.

Example 1

Solve $-3 < \dfrac{-2x-7}{5} < 7$.

Multiply each part to remove the denominator from the middle expression: 

$-3\cdot (5) < \dfrac{-2x-7}{5} \cdot (5) < 7 \cdot (5)$

$-15 < -2x-7 < 35$

Isolate $x$ in the middle of the inequality:

$- 15 + 7 < -2x -7 + 7 < 35 + 7$

$- 8 < -2x < 42$

Now divide each part by -2 (and remember to change the direction of the inequality symbol!):

$\displaystyle \frac{-8}{-2} > \frac{-2x}{-2} > \frac{42}{-2}$

$4 > x > -21 $

Finally, it is customary (though not necessary) to write the inequality so that the inequality arrows point to the left (i.e., so that the numbers proceed from smallest to largest):

$-21 < x < 4$