Chapter 5.3, Problem 15PPS

To determine

To solve the inequality:

  $\frac{w}{8}-13>-6$

And to graph the solution on a number line.

Answer to Problem 15PPS

  $\left(-\infty ,3\right)$

  

Explanation of Solution

Given:

  $\frac{w}{8}-13>-6$

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

  $\frac{w}{8}-13>-6$

First, add 13 both sides of the inequality, as

  $\begin{array}{c}\frac{w}{8}-13>-6\\ \Rightarrow \frac{w}{8}-13+13>-6+13\\ \Rightarrow \frac{w}{8}>7\end{array}$

Now, multiply both sides of the inequality with 8, as

  $\begin{array}{c}\frac{w}{8}>7\\ \Rightarrow \frac{w}{8}\times 8>7\times 8\\ \Rightarrow w>56\end{array}$

Thus, the solution of given inequality is every number strictly greater than 56, this can be represented in interval form, as

  $\left(56,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, 56, is strict, so use a hollow circle at 56. And the solution interval has $\infty$ at right endpoint which means every number strictly greater than 56 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: