Chapter 1.7, Problem 20E

To determine

To calculate: The solution of the given inequality, $3x+1\ge 2+x$, and graph the solution set.

The solution set of the given inequality, $3x+1\ge 2+x$, is $x\ge \frac{1}{2}$.

Calculation:

Consider the given inequality, $3x+1\ge 2+x$.

Subtract x 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 $a+c<b+c$.

$2x+1\ge 2$

Subtract 1 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 $a+c<b+c$.

$2x\ge 1$

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 $ac<bc$.

$x\ge \frac{1}{2}$

The solution set of the given inequality is the set of all real numbers that are equal to or greater than $\frac{1}{2}$ which can be denoted by $\left[\frac{1}{2},\infty \right)$.

Graph:

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

The bracket at $x=\frac{1}{2}$ means that the value at $x=\frac{1}{2}$ is included in the solution set of the given inequality.