Chapter 1.5, Problem 38PPSE

Solution

To solve the compound inequality $6x\ge -24\text{ and }9x<54$ and graph the solution.

  $-4\le x<6$

The graph of solution set is shown below:

  

Given:

The compound inequality, $6x\ge -24\text{ and }9x<54$ .

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 $6x\ge -24\text{ and }9x<54$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}6x\ge -24\text{and}9x<54\\ \frac{6x}{6}\ge \frac{-24}{6}\text{and}\frac{9x}{9}<\frac{54}{9}\\ x\ge -4\text{and}x<6\\ \Rightarrow -4\le x<6\end{array}$

Thus, the solution of the given inequality is

  $-4\le x<6$ .

Since the inequality sign is not strict with left endpoint, use the solid circles at the left endpoint. But the inequality sign is strict with the right endpoint, so use the hollow circle at right endpoint and shade the region between them as shown below