Chapter 5.4, Problem 10PPS

To determine

To solve below compound inequality:

  $-5<3p+7\le 22$

And to graph the solution on a number line.

Answer to Problem 10PPS

  $\left(-4,5\right]$

  

Explanation of Solution

Given:

  $-5<3p+7\le 22$

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:

  $-5<3p+7\le 22$

Subtract 7 from each side of the inequality, as

  $\begin{array}{c}-5<3p+7\le 22\\ \Rightarrow -5-7<3p+7-7\le 22-7\\ \Rightarrow -12<3p\le 15\end{array}$

To simplify further divide each side by 3, as

  $\begin{array}{c}-12<3p\le 15\\ \Rightarrow -\frac{12}{3}<\frac{3p}{3}\le \frac{15}{3}\\ \Rightarrow -4<p\le 5\end{array}$

Thus, the solution of given inequality is every number strictly greater than $-4$ but less than and equals to 5, this can be represented in interval form, as

  $\left(-4,5\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point, 5, is not strict, so use a solid circle at 5. And the solution interval at left endpoint, $-4$ , is strict so use a hollow circle at $-4$ . The graph of the solution of inequality is shown below: