Chapter 5.3, Problem 21PPS

To determine

To solve the inequality:

  $15h+30<10h-45$

And to graph the solution on a number line.

Answer to Problem 21PPS

  $\left(-\infty ,-15\right)$

  

Explanation of Solution

Given:

  $15h+30<10h-45$

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. $b<c\Rightarrow b\pm a<c\pm a$
2. $b<c\Rightarrow ab<ac,\text{ if }a>0$
3. $b<c\Rightarrow ab>ac,\text{ if }a<0$

Calculation:

In order to solve the inequality

  $15h+30<10h-45$

First, subtract 6 from both sides of the inequality, as

  $\begin{array}{c}15h+30<10h-45\\ \Rightarrow 15h+30-30<10h-45-30\\ \Rightarrow 15h<10h-75\\ \Rightarrow 15h-10h<10h-75-10h\\ \Rightarrow 5h<-75\end{array}$

Now, divide both sides of the inequality by 5, as

  $\begin{array}{c}5h<-75\\ \Rightarrow \frac{5h}{5}<\frac{-75}{5}\\ \Rightarrow h<-15\end{array}$

Thus, the solution of given inequality is every number strictly less than $-15$ , this can be represented in interval form, as

  $\left(-\infty ,-15\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point, $-15$ , is strict, so use a hollow circle at $-15$ . And the solution interval has $-\infty$ at left endpoint which means every number strictly less than $-15$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: