Chapter 5.4, Problem 8PPS

To determine

To solve the compound inequality:

  $y-1\ge 7,\text{ and }y+3<-1$

And to graph the solution on a number line.

Answer to Problem 8PPS

No solution.

Explanation of Solution

Given:

  $y-1\ge 7,\text{ and }y+3<-1$

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:

  $y-1\ge 7,\text{ and }y+3<-1$

First, solve the inequality

  $y-1\ge 7$

By adding 1 both sides, as

  $\begin{array}{c}y-1+1\ge 7+1\\ \Rightarrow y-0\ge 8\\ \Rightarrow y\ge 8\end{array}$

Now, solve the inequality

  $y+3<-1$

By subtracting 3 from both sides, as

  $\begin{array}{c}y+3-3<-1-3\\ \Rightarrow y+0<-4\\ \Rightarrow y<-4\end{array}$

Combining both of solutions:

  $y<-4,\text{ and }y\ge 8$

But there does not exist any number y which is less than $-4$ but greater than 8.

Thus, the given compound inequality has no solution.