Chapter 5.2, Problem 36PPS

To determine

To solve the inequality:

  $25j\ge 8$

And to graph the solution on a number line.

  

Answer to Problem 36PPS

The solution of the inequality is $\left[\frac{8}{25},\infty \right)\text{ or }j\ge 0.32$ and corresponding correct option is a.

Explanation of Solution

Given:

  $25j\ge 8$

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:

  $25j\ge 8$

Divide both sides by 25:

  $\begin{array}{c}25j\ge 8\\ \frac{25j}{25}\ge \frac{8}{25}\\ j\ge \frac{8}{25}\\ j\ge \text{0}\text{.32}\end{array}$

Thus, the inequality is true for h greater than $0.32$ , thus the solution of given inequality in interval form is:

  $\left(0.32,\infty \right)$

Now in order to sketch the graph of the solution of the inequality, since the interval of solution at left end point $0.32$ is strict, so use adark circle at $0.32$ . And the solution interval has $\infty$ at right endpoint which means every number greater than from $0.32$ to infinity is in the solution interval and does not contain the left end point in the number line. The graph of the inequality is shown below:

  

Hence, option a is correct.