Chapter 5.4, Problem 13PPS

To determine

To solve the compound inequality:

  $22\ge 4m-2,\text{ or }5-3m\le -13$

And to graph the solution on a number line.

Answer to Problem 13PPS

  $m=6$

  

Explanation of Solution

Given:

  $22\ge 4m-2,\text{ or }5-3m\le -13$

Concept Used:

• 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.
• Properties of inequality:

1. $b\le c\Rightarrow b\pm a\le c\pm a$
2. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$
3. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

In order to solve the compound inequality:

  $22\ge 4m-2,\text{ or }5-3m\le -13$

First, solve the inequality

  $22\ge 4m-2$

First, by adding 2 both sides, as

  $\begin{array}{c}22+2\ge 4m-2+2\\ \Rightarrow 24\ge 4m\end{array}$

Now, divide both sides by 4, as

  $\begin{array}{c}\frac{24}{4}\ge \frac{4m}{4}\\ \Rightarrow 6\ge m\end{array}$

Now, solve the inequality

  $5-3m\le -13$

First, by subtracting 5 from both sides, as

  $\begin{array}{c}5-3m-5\le -13-5\\ \Rightarrow -3m\le -18\end{array}$

Now, divide both sides of above inequality by $-3$ , as

  $\begin{array}{c}-\frac{3m}{-3}\ge -\frac{18}{-3}\\ \Rightarrow m\ge 6\end{array}$

Combining both of solutions:

  $\begin{array}{c}m\le 6\text{ or }m\ge 6\\ \Rightarrow m=6\end{array}$

Thus, the inequality is true only if the value of m is 6.

The graph of the solution of inequality, that is point 6 is shown below: