Chapter 5.3, Problem 17PPS

To determine

To solve the inequality:

  $37<7-10w$

And to graph the solution on a number line.

Answer to Problem 17PPS

  $\left(-3,\infty \right)$

  

Explanation of Solution

Given:

  $37<7-10w$

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

  $37<7-10w$

First, subtract 7 from both sides of the inequality, as

  $\begin{array}{c}37<7-10w\\ \Rightarrow 37-7<7-10w-7\\ \Rightarrow 30<-10w\end{array}$

Now, divide both sides of the inequality by $-10$ , that would change the inequality, as

  $\begin{array}{c}30<-10w\\ \Rightarrow \frac{30}{-10}>-\frac{10w}{-10}\\ \Rightarrow -3<w\end{array}$

Thus, the solution of given inequality is every number strictly greater than $-3$ , this can be represented in interval form, as

  $\left(-3,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-3$ , is strict, so use a hollow circle at $-3$ . And the solution interval has $\infty$ at right endpoint which means every number strictly greater than $-3$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: