Chapter 5.1, Problem 68SR

To determine

To solve the inequality:

  $\frac{1}{4}m\le -17$

And to graph the solution on a number line.

Answer to Problem 68SR

  $\left(-\infty ,-68\right]$

  

Explanation of Solution

Given:

  $\frac{1}{4}m\le -17$

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. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

To solve the inequality:

  $\frac{1}{4}m\le -17$

Multiply both sides of the inequality with 4, as

  $\begin{array}{c}\frac{1}{4}m\le -17\\ 4\times \frac{1}{4}m\le 4\times \left(-17\right)\\ m\le -68\end{array}$

Thus, the inequality is true for m less than $-68$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,-68\right]$

Now in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-68$ is not strict, so use a solid circle at $-68$ . And the solution interval has $-\infty$ at left endpoint which means every number less than $-68$ to negative infinity is in the solution interval. The graph of the inequality is shown below: