Chapter 5.3, Problem 14PPS

To determine

To solve the inequality:

  $-9\ge \frac{2}{3}m+7$

And to graph the solution on a number line.

Answer to Problem 14PPS

  $\left(-\infty ,-24\right]$

  

Explanation of Solution

Given:

  $-9\ge \frac{2}{3}m+7$

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 inequality

  $-9\ge \frac{2}{3}m+7$

First, subtract 7 from both sides of the inequality and simplify further using properties of the inequality, as

  $\begin{array}{c}-9\ge \frac{2}{3}m+7\\ \Rightarrow -9-7\ge \frac{2}{3}m+7-7\\ \Rightarrow -16\ge \frac{2}{3}m+0\\ \Rightarrow -16\times 3\ge \frac{2}{3}m\times 3\\ \Rightarrow -48\ge 2m\\ \Rightarrow -\frac{48}{2}\ge \frac{2m}{2}\\ \Rightarrow -24\ge m\end{array}$

Thus, the solution of given inequality is every number less than and equals to $-24$ , this can be represented in interval form, as

  $\left(-\infty ,-24\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-24$ is not strict, so use a solid circle at $-24$ . And the solution interval has $-\infty$ at left endpoint which means every number less than and equals to $-24$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: