Chapter 1.7, Problem 38E

To determine

To calculate: The solution of the inequality $-1\le \frac{-3x+5}{7}\le 2$ and to graph the solution set.

The solution set of the inequality $-1\le \frac{-3x+5}{7}\le 2$ is $-3\le x\le 4$.

Calculation:

The given inequality is $-1\le \frac{-3x+5}{7}\le 2$.

Multiply each part by 7 using the multiplication property of the inequality, which says that if $c>0$, then $a<b$ becomes $ac<bc$ and if $c<0$, then $a>b$ becomes $ac<bc$.

$-7\le -3x+5\le 14$

Subtract 5 from each part using the addition of the constant property of inequality, which says that if $a<b$, then $a<b$ becomes $a+c<b+c$.

$-12\le -3x\le 9$

Divide each part by $-3$ using the multiplication property of the inequality, which says that if $c>0$, then $a<b$ becomes $ac<bc$ and if $c<0$, then $a>b$ becomes $ac<bc$.

$\begin{array}{l}4\ge x\ge -3\\ -3\le x\le 4\end{array}$

The solution set of the inequality is all real numbers, which are greater than or equal to $-3$ and less than or equal to $4$, denoted by $\left[-3,4\right]$.

Graph:

The graph of the solution set is shown as:

The bracket at $x=-3$ and $x=4$ means that the point is included in the solution set.