Vicki volunteers for 3-hour shifts at the library on the weekends. During the week, she volunteers for 2-hour shifts at a nursing home. She volunteers for more than 24 hours a week.

Question

What graph represents that situation?

Answer 1

To represent Vicki's volunteering schedule graphically, we can set up a coordinate system where one axis represents the number of hours she volunteers at the library during the weekend and the other represents the hours she volunteers at the nursing home during the week.

Let's define:

• Let \( x \) be the number of 3-hour shifts at the library she works on the weekends.
• Let \( y \) be the number of 2-hour shifts at the nursing home during the week.

From the problem, we know that:

• Each shift at the library is 3 hours, so the total hours she volunteers at the library is \( 3x \).
• Each shift at the nursing home is 2 hours, so the total hours she volunteers at the nursing home is \( 2y \).
• Vicki volunteers for more than 24 hours a week, which gives us the inequality:

\[ 3x + 2y > 24 \]

To graph this inequality:

1. Convert the inequality into an equation:

   • The line representing the boundary of the inequality can be found by setting \( 3x + 2y = 24 \).

2. Find intercepts:

   • To find the x-intercept (where \( y = 0 \)): \[ 3x + 2(0) = 24 \implies x = 8. \]
   • To find the y-intercept (where \( x = 0 \)): \[ 3(0) + 2y = 24 \implies y = 12. \]

3. Plot the line:

   • Plot the points (8, 0) and (0, 12) on the graph, and draw a dashed line through these points to indicate that the boundary itself is not included (since it's a "greater than" inequality).

4. Shade the appropriate area:

   • Since we want \( 3x + 2y > 24 \), shade the region above the line, which represents all combinations of \( x \) and \( y \) that satisfy the inequality.

This graph visualizes the different combinations of shifts Vicki can work at the library and the nursing home to total more than 24 hours a week.