Chapter 5.2, Problem 55SR

To determine

To solve the inequality:

  $-\frac{4}{3}p>-10$

And to graph the solution on a number line.

Answer to Problem 55SR

  $\left(-\infty ,7.5\right)$

  

Explanation of Solution

Given:

  $-\frac{4}{3}p>-10$

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. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$

Calculation:

To solve the inequality:

  $-\frac{4}{3}p>-10$

Multiply both sides of the inequality by $-1$ to reverse the inequality:

  $\begin{array}{c}\left(-\frac{4}{3}p\right)\left(-1\right)<\left(-10\right)\left(-1\right)\\ \frac{4}{3}p<10\end{array}$

Multiply both sides of the inequality by $3$ and simplify:

  $\begin{array}{c}3\cdot \frac{4}{3}p<10\cdot 3\\ 4p<30\\ \frac{4p}{4}<\frac{30}{4}\\ p<\frac{15}{2}\end{array}$

Thus, the inequality is true for m less than $\frac{15}{2}$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,\frac{15}{2}\right]\text{}\text{or }\left(-\infty ,7.5\right]$

Now in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $7.5$ is not strict, so use a blank circle at $7.5$ . And the solution interval has $-\infty$ at left endpoint which means every number less than $7.5$ to negative infinity is in the solution interval. The graph of the inequality is shown below: