Chapter 3.3, Problem 42E

To determine

To solve the compound inequality:

$4v+9\le 5,\text{ or }-3v\ge -6$

And to graph the solution on a number line.

Answer to Problem 42E

  $\left(-\infty ,-1\right]$

Explanation of Solution

Given:

$4v+9\le 5,\text{ or }-3v\ge -6$

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:

$4v+9\le 5,\text{ or }-3v\ge -6$

First, solve the inequality

$4v+9\le 5$

By subtracting 9 from both sides and then using properties of inequality to simplify further, as

  $\begin{array}{c}4v+9\le 5\\ \Rightarrow 4v+9-9\le 5-9\\ \Rightarrow 4v\le -4\\ \Rightarrow 4v\le -\frac{4}{4}\\ \Rightarrow v\le -1\end{array}$

Now, solve the inequality

$-3v\ge -6$

By dividing both sides by $-3$ that would change the above inequality, as

  $\begin{array}{c}-3v\ge -6\\ \Rightarrow -\frac{3v}{-3}\le \frac{-6}{-3}\\ \Rightarrow v\le 2\end{array}$

Combining both of solutions:

  $v\le 2,\text{ and }v\le -1\Rightarrow v\le -1$

Thus, the solution of given inequality is every number less than and equals to $-1$ , this can be represented in interval form, as

  $\left(-\infty ,-1\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point $-1$ is not strict. And the solution interval at left endpoint $-\infty$ . Thus, the solution of given inequality is every number less than and equals to $-1$ . The graph of the solution of inequality is shown below: