Chapter 2.6, Problem 8E

To determine

To find: The solution of the inequality $\left|4c+5\right|>7$ and graph the solution if possible.

Answer to Problem 8E

  $\left(-\infty ,-3\right)\cup \left(\frac{1}{2},\infty \right)$

  

Explanation of Solution

Given information:

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

Concept Used:

Absolute value of a number:

The absolute value of a number ‘ $a$ ’ is equal to the distance that ‘ $a$ ’ is from zero.

  $\left|a\right|=\{\begin{array}{c}a,\text{if}a\ge 0,\text{and}\\ -a,\text{if}a<0\end{array}$

Rules for absolute Value inequalities:

If $\left|p\right|>a$ , $a>0$ then $p<-a$ or $u>a$ .

If $\left|p\right|<a$ , $a<0$ then $-a<p<a$ .

If $\left|p\right|\ge a$ , $a>0$ then $p\le -a$ or $p\ge a$ .

If $\left|p\right|\le a$ , $a>0$ then $-a\le p\le a$ .

Calculation:

Use the rule of absolute inequalities: If $\left|p\right|>a$ , $a>0$ then $p<-a$ or $u>a$ .

Case 1:

  $4c+5<-7$

Subtract $5$ from both the sides.

  $\begin{array}{c}4c+5-5<-7-5\\ 4c<-12\end{array}$

Divide by $4$ on both the sides.

  $\begin{array}{c}\frac{4c}{4}<\frac{-12}{4}\\ c<-3\end{array}$

Case 2:

  $4c+5>7$

Subtract $5$ from both the sides.

  $\begin{array}{c}4c+5-5>7-5\\ 4c>2\end{array}$

Divide by $4$ on both the sides.

  $\begin{array}{c}\frac{4c}{4}>\frac{2}{4}\\ c>\frac{1}{2}\end{array}$

Graph:

  

From the above graph, it is observed that $x$ represents all real number that are less than $-3$ or greater than $\frac{1}{2}$ .

Therefore, the solution of the given inequality is $\left(-\infty ,-3\right)\cup \left(\frac{1}{2},\infty \right)$ .