Chapter 1.7, Problem 26E

To determine

To calculate: The solution of the given inequality, $3+\frac{2x}{7}>x-2$, and graph the solution set.

The solution set of the given inequality, $3+\frac{2x}{7}>x-2$, is $x<7$.

Calculation:

Consider the given inequality, $3+\frac{2x}{7}>x-2$.

Multiply each part by 7 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 $ac<bc$.

$21+2x>7x-14$

Subtract $7x$ and 21 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 $a+c<b+c$.

$-5x>-35$

Divide each part by $-5$ 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 $ac<bc$.

$x<7$

The solution set of the inequality is the set of all real numbers that are less than 7 which can be denoted by $\left(-\infty ,7\right)$.

Graph:

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

The parenthesis at $x=7$ means that the value at $x=7$ is not included in the solution set of the given inequality.