Chapter 5.2, Problem 27PPS

To determine

To solve the inequality:

  $4b\le -3$

And to graph the solution on a number line.

Answer to Problem 27PPS

  $\left(-\infty ,-0.75\right]$

  

Explanation of Solution

Given:

  $4b\le -3$

Concept Used:

• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.endpoint.
• To graph endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Properties of inequality:
1. $b\le c\Rightarrow ab\le ac,\text{ if }a>0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a<0$

Calculation:

To solve the inequality:

  $4b\le -3$

Divide both sides of the inequality by 4, as

  $\begin{array}{c}4b\le -3\\ \Rightarrow \frac{4b}{4}\le -\frac{3}{4}\\ \Rightarrow b\le -\frac{3}{4}\\ \Rightarrow b\le -0.75\end{array}$

Thus, the solution of given inequality is every number less than and equals to $-0.75$ , this can be represented in interval form, as

  $\left(-\infty ,-0.75\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-0.75$ is not strict, so use a solid circle at $-0.75$ . And the solution interval has $-\infty$ at left endpoint which means every number less than and equals to $-0.75$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: