Chapter 1.7, Problem 88E

To determine

To solve: the absolute value inequality. Express the solution using interval notation and graph the solution set.

Answer to Problem 88E

  $x>\frac{-15}{7}orx<\frac{-13}{7}$ .

Explanation of Solution

Given:

The given inequality is $7\left|x+2\right|+5>4$ .

Concept used:

Properties of absolute value inequalities:

Inequality	Equivalent form	Graph
$\left|x\right|<c$	$-c<x<c$
$\left|x\right|\le c$	$-c\le x\le c$
$\left|x\right|>c$	$x<-c\text{ or }c<x$
$\left|x\right|\ge c$	$x\le -c\text{ or }c\le x$

Calculation:

The given expression is expressed as

  $\begin{array}{l}7\left|x+2\right|>-1\left\{\text{subtract 5 from both the sides}\right\}\\ \Rightarrow \left|x+2\right|>\frac{-1}{7}\left\{\text{divide 7 from both the sides}\right\}\end{array}$

According to the properties of absolute value inequalities, thegiven inequalityis expressed as

  $\begin{array}{l}\left|x+2\right|>\frac{-1}{7}\\ \therefore x+2>\frac{-1}{7}orx+2<-\left(\frac{-1}{7}\right)\\ x+2>\frac{-1}{7}orx+2<\frac{1}{7}\\ \{\text{subtract 2 on both the sides}}\\ \Rightarrow x>\frac{-1}{7}-2orx<\frac{1}{7}-2\\ \therefore x>\frac{-15}{7}orx<\frac{-13}{7}\end{array}$

Hence,the solution set is $x>\frac{-15}{7}orx<\frac{-13}{7}$ .

The solution set of the inequality graphed on the number line.