Chapter 1.6, Problem 40PPSE

Solution

To write the compound inequality $50\le b\le 55$ as an absolute value inequality.

  $\left|b-52.5\right|\le 2.5$

Given:

The compound inequality $50\le b\le 55$ .

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: $\text{if }\left|x\right|<a,\text{then}-a<x<a$

Calculation:

In order to write the compound inequality $50\le b\le 55$ as an absolute value inequality, note that the midpoint of $50\text{ and 55}$ is $52.5$ . And each endpoint is $2.5$ unit from $52.5$ .

Thus, the given inequality can be written as an absolute value inequality as $\left|b-52.5\right|\le 2.5$ .