Chapter 3.3, Problem 40E

To determine

To solve the compound inequality:

$2d+7<-9,\text{ or }4d-1>-3$

And to graph the solution on a number line.

Answer to Problem 40E

No solution.

Explanation of Solution

Given:

$2d+7<-9,\text{ or }4d-1>-3$

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:

$2d+7<-9,\text{ or }4d-1>-3$

First, solve the inequality

$2d+7<-9$

First, by subtracting 7 from both sides and then using properties of inequality to simplify further, as

  $\begin{array}{c}2d+7-7<-9-7\\ \Rightarrow 2d<-16\\ \Rightarrow d<\frac{-16}{2}\\ \Rightarrow d<-8\end{array}$

Now, solve the inequality

$4d-1>-3$

First by adding 1 both sides and then using properties of inequality to simplify further, as

  $\begin{array}{c}4d-1+1>-3+1\\ \Rightarrow 4d>-2\\ \Rightarrow d>\frac{-2}{4}\\ \Rightarrow d>-0.5\end{array}$

Combining both of solutions:

  $d<-8,\text{ and }d>-0.5$

Now, $-8<-0.5$ .

Thus, there does not exist any number d which is strictly less than $-8$ but strictly greater than $-0.5$ .

Thus, the given compound inequality has no solution.