Chapter 1.5, Problem 37PPSE

Solution

To solve the compound inequality $3x\ge -12\text{ and }8x\le 16$ and graph the solution.

  $-4\le x\le 2$

The graph of solution set is shown below:

  

Given:

The compound inequality, $3x\ge -12\text{ and }8x\le 16$ .

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.

Calculation:

In order to solve the given inequality $3x\ge -12\text{ and }8x\le 16$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}3x\ge -12\text{ and }8x\le 16\\ \frac{3x}{3}\ge \frac{-12}{3}\text{ and }\frac{8x}{8}\le \frac{16}{8}\\ x\ge -4\text{ and }x\le 2\\ \Rightarrow -4\le x\le 2\end{array}$

Thus, the solution of the given inequality is $-4\le x\le 2$ .

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