Chapter 1.7, Problem 40E

To determine

To calculate: The solution of the inequality $-1<2-\frac{x}{3}<1$ and graph the solution set.

The solution set of the inequality $-1<2-\frac{x}{3}<1$ is $3<x<9$.

Calculation:

The given inequality is $-1<2-\frac{x}{3}<1$.

Subtract 2 from each part using the addition of the constant property of the inequality, which says that if $a<b$, then $a<b$ becomes $a+c<b+c$.

$-3<-\frac{x}{3}<-1$

Multiply 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 $ac<bc$.

$\begin{array}{l}9>x>3\\ 3<x<9\end{array}$

So, the solution set of the inequality is all real numbers, which are greater than $3$ and less than 9, denoted by $\left(3,9\right)$.

Graph:

The graph of the solution set is shown as:

The parenthesis at $x=3$ and $x=9$ means that the point is not included in the solution set.