Chapter 5.4, Problem 55SR

To determine

To solve the inequality:

  $-\frac{5}{4}p+6<12$

And to graph the solution on a number line.

Answer to Problem 55SR

  $\left(-4.8,\infty \right)$

  

Explanation of Solution

Given:

  $-\frac{5}{4}p+6<12$

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

  $-\frac{5}{4}p+6<12$

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

  $\begin{array}{c}-\frac{5}{4}p+6<12\\ \Rightarrow -\frac{5}{4}p+6-6<12-6\\ \Rightarrow -\frac{5}{4}p<6\end{array}$

Now, multiply both sides of the inequality with 4, and then again use properties of the inequalities to solve it further, as

  $\begin{array}{c}-\frac{5}{4}p<6\\ \Rightarrow -\frac{5}{4}p\times 4<6\times 4\\ \Rightarrow -5p<24\\ \Rightarrow -\frac{5p}{-5}>\frac{24}{-5}\\ \Rightarrow p>-4.8\end{array}$

Thus, the solution of given inequality is every number strictly greater than $-4.8$ , this can be represented in interval form, as

  $\left(-4.8,\infty \right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point, $-4.8$ , is strict, so use a hollow circle at $-4.8$ . And the solution interval has $\infty$ at right endpoint which means every number strictly greater than $-4.8$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: