Chapter 1.6, Problem 86PPSE

Solution

To graph the solution $\left|x\right|\ge 6\text{ or }\left|x\right|<5$ .

The graph of solution set is:

  

Given:

The inequality, $\left|x\right|\ge 6\text{ or }\left|x\right|<5$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$ , use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$ , use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.
• The absolute value property: $\text{if }\left|x\right|<a,\text{then}-a<x<a$
• The absolute value property: $\text{if }\left|x\right|>a,\text{then}x>a\text{ or }x<-a$

Calculation:

In order to graph the given compound inequality $\left|x\right|\ge 6\text{ or }\left|x\right|<5$ , use the absolute inequality property and simplify further as shown below

  $\begin{array}{l}\left|x\right|\ge 6\text{or }\left|x\right|<5\\ x\ge 6\text{or}x\le -6\text{ and }-5<x<5\\ \Rightarrow x\le -6\text{ or }-5<x<5\text{or }x\ge 6\end{array}$

The solution region is shown below: