Question

Jeanne babysits for $6 per hour. She also works as a reading tutor for $10 per hour. She is only allowed to work 20 hours per week. This week, her goal is to make at least $75.

A. Use a system of inequalities to model the scenario above. Let x represent babysitting hours and y represent tutoring hours.
B. Use the model created in part A to create a graph representing Jeanne’s probable income earned and possible number of hours worked this week.
C. Analyze the set of coordinate values that represent solutions for the model created in part A. Choose one of the coordinates within the solution and algebraically prove that the coordinate represents a true solution for the model.

Answer 1

x+y≤20

6x+10y ≥ 75

Explanation:

Let the number of hours of babysitting be x and that for tutoring be y

As per the given conditions

The total no of hours of work must be less than or equal to 20

Hence

x+y≤20

Also Her target is to earn atleast $75

Hence the second inequation will be

6x+10y≥75

Hence our system of inequatlites representing above conditions are

x+y≤20

6x+10y≥75

Now in order to graph them , we first graph the lines x+y=20 and 6x+10y=75 and shade the region which satisfies the respective inequality by taking a coordinate (0,0) .

Please refer to the graph attached with this.

The shaded region gives us the set of coordinates probably the solution to above inequations.

Let us pick one coordinate (10,5) from the shaded region and check for the solution.

put (10,5) in two inequations and see if they are true for them.

10+5≤20

15≤20 True

6(10)+10(5) ≥75

60+50≥75

110≥75 true

Hence checked , both stands true for (10,5)

Answer 2

For the answer to the question above, I would start with a simple equation 6x+10y is more than or equal to 75
A.
If we let x represent babysitting hours and y represent tutoring hours:
x+ y ≤ 20
6x + 10y ≥ 75

B. The inequality: x+ y ≤ 20 can be graphed by graphing the line x + y = 20 and shading the area below the line.
The inequality 6x + 10y ≥ 75 can be graphed by graphing the line 6x + 10y = 75 and shading the area above the line.

C. The area where the two shaded regions from the two inequalities overlap are the possible number of hours for tutoring and for babysitting. Algebraically:
x ≤ 20 - y
6(20 - y) +10y ≥ 75
y ≤ 11.25
x ≤ 8.75