Chapter 1, Problem 33E

Solution

To solve the inequality $\left|7x\right|+3\le 21$ and graph the solution.

  $-\frac{18}{7}\le x\le \frac{18}{7}$

The graph of solution is shown below:

  

Given:

The inequality, $\left|7x\right|+3\le 21$ .

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: $\text{if }\left|x\right|>a,\text{then}x>a\text{ or }x<-a$
• The absolute value property: $\text{if }\left|x\right|<a,\text{then}-a<x<a$

Calculation:

In order to solve the given inequality $\left|7x\right|+3\le 21$, use the absolute value property and isolate the variable on one side by performing the basic algebraic operations as shown below

  $\begin{array}{c}\left|7x\right|+3\le 21\\ \left|7x\right|+3-3\le 21-3\\ \left|7x\right|\le 18\\ -18\le 7x\le 18\\ -\frac{18}{7}\le \frac{7x}{7}\le \frac{18}{7}\\ -\frac{18}{7}\le x\le \frac{18}{7}\end{array}$

Thus, the solution of the inequality is $-\frac{18}{7}\le x\le \frac{18}{7}$ .

Since the inequality sign is not strict with both endpoints, use the solid circles at both endpoints and shade the region between them as shown below