Chapter 5.2, Problem 53SR

To determine

To solve the inequality:

  $-7f>5$

And to graph the solution on a number line.

Answer to Problem 53SR

  $\left(-\infty ,-\frac{5}{7}\right)$

  

Explanation of Solution

Given:

  $-7f>5$

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

Calculation:

To solve the inequality:

  $-7f>5$

Divide both sides of the inequality by $-7$ which would change sign of the inequality, as

  $\begin{array}{c}-7f>5\\ \Rightarrow -\frac{7f}{-7}<\frac{5}{-7}\\ \Rightarrow f<-\frac{5}{7}\end{array}$

Thus, the solution of given inequality is every number strictly less than $-\frac{5}{7}$ , this can be represented in interval form, as

  $\left(-\infty ,-\frac{5}{7}\right)$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-\frac{5}{7}$ is strict, so use a hollow circle at $-\frac{5}{7}$ . And the solution interval has $-\infty$ at left endpoint which means every number strictly less than $-\frac{5}{7}$ is in the solution interval of given inequality. The graph of the solution of inequality is shown below: