Chapter 5.2, Problem 38PPS

To determine

To solve the inequality:

  $2.3<-5t$

Answer to Problem 38PPS

The solution of the inequality is $\left(-\infty ,-0.46\right)$ and corresponding correct option is c.

  

Explanation of Solution

Given:

The graph is given.

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:

  $2.3<-5t$

Multiply both sides by -1 to reverse the inequality:

  $\begin{array}{c}2.3<-5t\\ 2.3\left(-1\right)>\left(-5t\right)\left(-1\right)\\ -2.3>5t\end{array}$

Divide both sides by 5:

  $\begin{array}{l}-2.3>5t\\ \frac{-2.3}{5}>\frac{5t}{5}\\ -0.46>t\end{array}$

Thus, the inequality is true for tless than $-0.46$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,-0.46\right)$

Now in order to sketch the graph of the solution of the inequality, since the interval of solution at right end point $-0.46$ is strict, so use a blank circle at $-0.46$ . And the solution interval has $-\infty$ at left endpoint which means every number less than from $-0.46$ to negative infinity is in the solution interval and does not contain the right end point in the number line. The graph of the inequality is shown below:

  

Hence, option c is correct.