Chapter 5.4, Problem 54SR

To determine

To solve the inequality:

  $8-\frac{z}{3}\ge 11$

And to graph the solution on a number line.

Answer to Problem 54SR

  $\left(-\infty ,-9\right]$

  

Explanation of Solution

Given:

  $8-\frac{z}{3}\ge 11$

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

  $8-\frac{z}{3}\ge 11$

First, subtract 8 from both sides of the inequality, as

  $\begin{array}{c}8-\frac{z}{3}\ge 11\\ \Rightarrow 8-\frac{z}{3}-8\ge 11-8\\ \Rightarrow 0-\frac{z}{3}\ge 3\\ \Rightarrow -\frac{z}{3}\ge 3\end{array}$

Now, multiply both side of the inequality with $-3$ that would change the inequality as

  $\begin{array}{c}-\frac{z}{3}\ge 3\\ \Rightarrow -\frac{z}{3}\times -3\le 3\times -3\\ \Rightarrow z\le -9\end{array}$

Thus, the solution of given inequality is every number less than and equals to $-9$ , this can be represented in interval form, as

  $\left(-\infty ,-9\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point $-9$ is not strict, so use a solid circle at $-9$ . And the solution interval has $-\infty$ at right endpoint which means every number less than and equals to $-9$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: