Chapter 3.3, Problem 37E

To determine

To solve below compound inequality:

$-2\le x-11\le 6$

And to graph the solution on a number line.

Answer to Problem 37E

  $\left[9,17\right]$

Explanation of Solution

Given:

$-2\le x-11\le 6$

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Properties of inequality:

1. $b\le c\Rightarrow b\pm a\le c\pm a$

2. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$

3. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

In order to solve the compound inequality:

$-2\le x-11\le 6$

Add 11 each side of the inequality, as

  $\begin{array}{c}-2+11\le x-11+11\le 6+11\\ \Rightarrow 9\le x\le 17\end{array}$

Thus, the solution of given inequality is every number greater than and equals to 9 but less than and equals to 17, this can be represented in interval form, as

  $\left[9,17\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, 9, is not strict , and also at right endpoint, 17, is not strict, so use a solid circle at both end points 9 and 17. The graph of the solution of inequality is shown below: