Chapter 5.2, Problem 68S

To determine

The number of employees are there in the factory.

Answer to Problem 68S

There are less than 63 employees are there in the factory.

Explanation of Solution

Given:

About $2/3$ of the employees are less than 42.

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:

Let the total number of the employees are x, and about $2/3$ of the employees are less than 42.

Then from the above information:

  $\frac{2}{3}x<42$

Multiply both sides by 3 to the inequality:

  $\begin{array}{c}\frac{2}{3}x<42\\ 3\cdot \frac{2}{3}x<42\cdot 3\\ 2x<126\end{array}$

Divide both sides by 2:

  $\begin{array}{c}2x<126\\ \frac{2x}{2}<\frac{126}{2}\\ x<63\end{array}$

Thus, the inequality is true for xless than 63, thus the solution of given inequality in interval form is:

  $x<63$

Hence, there are less than 63 employees are there in the factory.