Chapter 6.6, Problem 5STP

A young woman wants to make at least $200 a week and can work no more than 30 hours a week. She works at the library for $8 an hour and babysits for $6 an hour.

a. What system of inequalities shows the possible combination of hours and jobs she can work?

b. Why did you exclude points to the left of the $y$-axis and below the $x$-axis?

To determine

To find: the system of inequalities that show the possible combination of hours and jobs, the woman can work

Answer to Problem 5STP

The system of inequalities is as follows:

  $\begin{array}{l}x+y\le 30\\ 8x+6y\ge 200\end{array}$

Here, $x$ denotes number of hours she works at the library and $y$ denotes number of hours she does babysitting

Explanation of Solution

Given information: A young woman wants to make at least $\$200$ a week and can work no more than $30$ hours a week. She works at the library for $\$8$ an hour and babysits for $\$6$ an hour.

First find the inequality regarding total number of hours she works in a week.

Let $x$ denotes number of hours she works at the library and $y$ denotes number of hours she do babysitting.

As she works for no more than $30$ hours a week,

  $x+y\le 30$

Now find the second inequality regarding total amount earned by her a week.

Amount earned by her per hour at the library $=\$8$

Amount earned by her in $x$ hours at the library $=\$8x$

Amount earned by her per hour from babysitting $=\$6$

Amount earned by her in $y$ hours from babysitting $=\$6y$

So,

Total amount earned by her $=\$8x+\$6y$

Also, it’s given that she wants to make at least $\$200$ a week.

So,

  $\$8x+\$6y\ge 200$

To determine

To explain: reason behind excluding points to the left of y-axis and below the x-axis.

Answer to Problem 5STP

Points to the left of y-axis and below the x-axis are excluded as number of hours cannot be negative.

Explanation of Solution

x-axis denotes number of hours she works at the library and y-axis denotes number of hours she does babysitting.

Here, number of hours she works at the library and number of hours she does babysitting cannot be negative.

As time cannot be negative, points to the left of y-axis and below the x-axis are excluded as number of hours cannot be negative.