Chapter 1.7, Problem 32E

To determine

To calculate: The solution of the given inequality, $-9\le -2x-7<5$, and graph the solution set.

The solution set of the given inequality, $-9\le -2x-7<5$, is $-6<x\le 1$.

Calculation:

Consider the given inequality, $-9\le -2x-7<5$.

Add 7 from each part by using the property of addition of a constant to an inequality, according to which, if $a<b$, then $a<b$ becomes $a+c<b+c$.

$-2\le -2x<12$

Divide each part by $-2$ by using the multiplicative property of an inequality, according to which, if $c>0$, then $a<b$ becomes $ac<bc$ and if $c<0$, then $a>b$ becomes $ac<bc$.

$-6<x\le 1$

The solution set of the given inequality is the set of all real numbers that are greater than $-6$ and less than or equal to 1 which can be denoted by $\left(-6,1\right]$.

Graph:

The solution set of the inequality is shown in the graph.

The parenthesis at $x=-6$ means that this point is not included in the solution set.

The bracket at $x=1$ means that this point is included in the solution set.