Chapter 2.6, Problem 39E

To determine

To explain:the reason of obtaining intersection and union sets of the given inequalities.

Answer to Problem 39E

Thus, the reason of obtaining intersection and union sets of the given inequalities are explained.

Explanation of Solution

Given:

  $\left|x\right|>5$ is the union of two sets.

  $\left|x\right|<5$ is the intersection of two sets.

Calculation:

Given the inequalities:

  $\left|x\right|<5$

  $\left|x\right|>5$

The absolute value inequality $\left|x\right|<5$ can be written as a compound inequality:

  $-5<x<5$

This means there are two inequalities which must both are verified.

  $x>-5$ And $x<5$

Find the solution set of each inequality

  $x\in (-5,\infty )$ and $x\in (-\infty ,5)$

Thefinal result is the intersection of the two sets:

  $\Rightarrow x\in (-5,\infty )\cap (-\infty ,5)=(-5,5)$

The absolute value inequality $\left|x\right|>5$ can be written:

  $x<-5$ or $x>5$

Because of dealing with an $\text{OR}$ issue, the final solutionis the reunion of the solution sets of the two inequalities:

  $x\in (-\infty ,-5)$ or $x\in (5,\infty )$

  $\Rightarrow x\in (-\infty ,-5)\cup (5,\infty )$

Conclusion:

Thus, the reason of obtaining intersection and union sets of the given inequalities are explained.