Chapter 9.4, Problem 70SR

To determine

Solve the inequality.

Answer to Problem 70SR

The solution set is $\{y|y>9\text{or}y<-5\}\text{or}(-\infty ,-5)\cup (9,\infty )$

Explanation of Solution

Given:

The inequality: $\left|y-2\right|>7$

Concept Used:

An absolute value inequality is an inequality that has an absolute value sign with a variable inside.

When solving absolute value inequalities, there are two cases to consider.

Case 1 : The expression inside the absolute value symbols is positive.

Case 2 : The expression inside the absolute value symbols is negative.

The solution is the intersection of the solutions of these two cases.

In other words, for any real numbers a and b , if |   a   | < b , then a < b AND a > - b

Rule:

  $\begin{array}{l}\text{If}\left|x\right|>a\Rightarrow x<-a\text{or}x>a\\ \text{If}\left|x\right|\ge a\Rightarrow x\le -a\text{or}x\ge a\end{array}$

  $\begin{array}{l}\text{If}\left|x\right|<a\Rightarrow -a<x<a\\ \text{If}\left|x\right|\le a\Rightarrow -a\le x\le a\end{array}$

Calculation:

  $\begin{array}{l}\left|y-2\right|>7\\ \text{If}\left|x\right|>a\Rightarrow x<-a\text{or}x>a\\ \left|y-2\right|>7\\ y-2<-7\text{or}y-2>7\\ y-2+2>-7+2\text{or}y-2+2>7+2\\ y<-5\text{or}y>9\end{array}$

The solution set is $\{y|y>9\text{or}y<-5\}\text{or}(-\infty ,-5)\cup (9,\infty )$

Thus, the solution set is $\{y|y>9\text{or}y<-5\}\text{or}(-\infty ,-5)\cup (9,\infty )$