Chapter 5.4, Problem 1AGP

To determine

To solve the compound inequality:

  $y-3\ge -11,\text{ and }y-3\le -8$

And to graph the solution on a number line.

Answer to Problem 1AGP

  $\left[-8,-5\right]$

  

Explanation of Solution

Given:

  $y-3\ge -11,\text{ and }y-3\le -8$

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:

  $y-3\ge -11,\text{ and }y-3\le -8$

First, solve the inequality

  $y-3\ge -11$

By adding 3 both sides, as

  $\begin{array}{c}y-3+3\ge -11+3\\ \Rightarrow y-0\ge -8\\ \Rightarrow y\ge -8\end{array}$

Now, solve the inequality

  $y-3\le -11$

By again adding 3 both sides, as

  $\begin{array}{c}y-3+3\le -8+3\\ \Rightarrow y-0\le -5\\ \Rightarrow y\le -5\end{array}$

Combining both of solutions:

  $-8\le y\le -5$

Thus, the solution of given inequality is in between $-8$ and $-5$ , this can be represented in interval form, as

  $\left[-8,-5\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left and right end points is not strict, so use solid circles at both the points. And the solution interval every number in between $-8$ and $-5$ . Thus, the graph of the solution of inequality is shown below: