Chapter 1.7, Problem 34E

To determine

To calculate: The solution of the given inequality, $-1\le -\left(x-4\right)<7$, and graph the solution set.

The solution set of the given inequality, $-1\le -\left(x-4\right)<7$, is $-3<x\le 5$.

Calculation:

Consider the given inequality, $-1\le -\left(x-4\right)<7$.

Multiply each part by $-1$ 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 $ac<bc$.

$1\ge \left(x-4\right)>-7$

Add 4 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 $a+c<b+c$.

$\begin{array}{c}5\ge x>-3\\ -3<x\le 5\end{array}$

The solution set of the given inequality is the set of all real numbers that are greater than $-3$ and less than or equal to $5$ which can be denoted by $\left(-3,5\right]$.

Graph:

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

The parenthesis at $x=-3$ means that this point is not included in the solution set.

The bracket at $x=5$ means that this point is included in the solution set.