Chapter 5.3, Problem 4CYU

To determine

To solve the inequality:

  $-3\le \frac{2}{3}r+9$

And to graph the solution on a number line.

Answer to Problem 4CYU

  $\left[-18,\infty \right)$

  

Explanation of Solution

Given:

  $-3\le \frac{2}{3}r+9$

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. $b<c\Rightarrow b\pm a<c\pm a$
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:

To solve the inequality:

  $-3\le \frac{2}{3}r+9$

First subtract 9 from both sides of the inequality and the simplify further using properties of the inequality, as

  $\begin{array}{c}-3\le \frac{2}{3}r+9\\ \Rightarrow -3-9\le \frac{2}{3}r+9-9\\ \Rightarrow -12\le \frac{2}{3}r+0\\ \Rightarrow -12\times 3\le \frac{2}{3}r\times 3\\ \Rightarrow -36\le 2r\\ \Rightarrow -\frac{36}{2}\le \frac{2r}{2}\\ \Rightarrow -18\le r\end{array}$

Thus, the solution of given inequality is every number greater than and equals to $-18$ , this can be represented in interval form, as

  $\left[-18,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point $-18$ is not strict, so use a solid circle at $-18$ . And the solution interval has $\infty$ at right endpoint which means every number greater than and equals to $-18$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: