Chapter 5.4, Problem 9PPS

To determine

To solve the compound inequality:

  $t+14\ge 15,\text{ and }t-9<-10$

And to graph the solution on a number line.

Answer to Problem 9PPS

No solution.

Explanation of Solution

Given:

  $t+14\ge 15,\text{ and }t-9<-10$

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:

  $t+14\ge 15,\text{ and }t-9<-10$

First, solve the inequality

  $t+14\ge 15$

By subtracting 14 from both sides, as

  $\begin{array}{c}t+14-14\ge 15-14\\ \Rightarrow t+0\ge 1\\ \Rightarrow t\ge 1\end{array}$

Now, solve the inequality

  $t-9<-10$

By adding 9 both sides, as

  $\begin{array}{c}t-9+9<-10+9\\ \Rightarrow t-0<-1\\ \Rightarrow t<-1\end{array}$

Combining both of solutions:

  $t<-1,\text{ and }t\ge 1$

But there does not exist any number y which is strictly less than $-1$ but greater than 1.

Thus, the given compound inequality has no solution.