Chapter 5.2, Problem 19PPS

To determine

To solve the inequality:

  $-\frac{3}{4}j\ge 12$

And to graph the solution on a number line.

Answer to Problem 19PPS

  $\left(-\infty ,16\right]$

  

Explanation of Solution

Given:

  $-\frac{3}{4}j\ge 12$

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

Calculation:

To solve the inequality:

  $-\frac{3}{4}j\ge 12$

First, multiply both sides of the inequality with $-4$ , which would change sign of the inequality, as

  $\begin{array}{c}-\frac{3}{4}j\ge 12\\ \Rightarrow \left(-4\right)\times \left(-\frac{3}{4}\right)j\le \left(-4\right)\times 12\\ \Rightarrow 3j\le 48\end{array}$

Now, divide both sides of inequality by 3, as

  $\begin{array}{c}-\frac{3}{4}j\ge 12\\ \Rightarrow 3j\le 48\\ \Rightarrow \frac{3j}{3}\le \frac{48}{3}\\ \Rightarrow j\le 16\end{array}$

Thus, the solution of given inequality is every number less than and equals to 16, this can be represented in interval form, as

  $\left(-\infty ,16\right]$

Now, in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point 16 is not strict, so use a solid circle at 16. And the solution interval has $-\infty$ at left endpoint which means every number less than and equals to 16 is in the solution interval of given inequality. The graph of the solution of inequality is shown below: