Chapter 5.6, Problem 61SR

To determine

To solve below compound inequality:

  $5n-1<-16,\text{ or }-3n-1<8$

And to graph the solution on a number line.

Answer to Problem 61SR

No solution.

Explanation of Solution

Given:

  $5n-1<-16,\text{ or }-3n-1<8$

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:

  $5n-1<-16,\text{ or }-3n-1<8$

First, simplify the inequality,

  $5n-1<-16$

For that first add 1 both sides and then simplify it further using properties of inequalities, as

  $\begin{array}{c}5n-1+1<-16+1\\ \Rightarrow 5n<-15\\ \Rightarrow \frac{5n}{5}<\frac{-15}{5}\\ \Rightarrow n<-3\end{array}$

Now, simplify the inequality

  $-3n-1<8$

For this too first add 1 both sides and then simplify it further using properties of inequalities, as

  $\begin{array}{c}-3n-1+1<8+1\\ \Rightarrow -3n<9\\ \Rightarrow -\frac{3n}{-3}>\frac{9}{-3}\\ \Rightarrow n>-3\end{array}$

Combining both of the above results, as

  $-3<n,\text{ and }n<-3$

But there does not exist any number that is greater than $-3$ and also is smaller than

  $-3$ .

Thus, the given inequality has no solution.