Chapter 12.3, Problem 1MRE

To determine

To calculate: The solution set for open sentence $\left|x-4\right|\le 7$ and graph it.

Answer to Problem 1MRE

The solution set is $\left[-3,11\right]$ . The graph is provided below,

Explanation of Solution

Given information:

The open sentence $\left|x-4\right|\le 7$ .

Formula used:

For an algebraic expression w and a real number n such that $n>0$ , absolute value inequality is $\left|w\right|\le n$ if and only if $-n\le w\le n$ .

Calculation:

Consider the open sentence $\left|x-4\right|\le 7$ .

Recall that for an algebraic expression w and a real number n such that $n>0$ , absolute value inequality is $\left|w\right|\le n$ if and only if $-n\le w\le n$ .

Apply it,

  $-7\le x-4\le 7$

Add 4 in each part of the above inequality,

  $\begin{array}{c}-7\le x-4\le 7\\ -7+4\le x-4+4\le 7+4\\ -3\le x\le 11\end{array}$

Write it in the interval form $\left[-3,11\right]$ .

The solution set is the closed interval that is all values of x that lie between $-3\text{ and 11}$ .

On the real number mark the points $x=-3\text{ and }x=\text{11}$ and darken the line between the points and the points also.

Thus, the solution set is $\left[-3,11\right]$ .