Tuesday, June 07, 2016

Daily Regents: Systems of Inequalities (August 2014)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams next week. At least, that is the plan.

August 2014, Questions 37

37. Edith babysits for x hours a week after school at a job that pays $4 an hour. She has accepted a job that pays $8 an hour as a library assistant working y hours a week. She will work both jobs. She is able to work no more than 15 hours a week, due to school commitments. Edith wants to earn at least $80 a week, working a combination of both jobs.
Write a system of inequalities that can be used to represent the situation. Graph these inequalities on the set of axes below.

Determine and state one combination of hours that will allow Edith to earn at least $80 per week while working no more than 15 hours.

If one job pays $4 per hour and she works x hours, she makes 4x dollars. If the other job pays $8 per hour and she works y hours, then she makes 5y dollars. The total is 4x + 8y, which must be greater than or equal to $80, so
4x + 8y > 80

If the total number of hours worked about both jobs must be less than or equal to 15 hours, then
x + y < 15

That is the system of inequalities to graph. Both lines will be solid. The one for her pay will be shaded above. The one for her hours will be shaded below.

You can graph these by finding the x- and y-intercepts, or by re-writing them in y-intercept form and putting them in the graphing calculator.

4x + 8(0) = 80
4x = 80
x = 20, (20, 0)
4(0) + 8y = 80
8y = 80
y = 10, (0, 10)
x + (0) = 15
x = 15, (15, 0)
(0) + y = 15
y = 15, (0, 15)

The answer to the second part varies. You can pick any part in the double-shaded region, S. Since the lines are solid, those boundary points are good as well.
So 0 hours babysitting and 10 hours at the library, or 3 hours babysitting and 9 hours at the library.

See image below.

Any questions?

If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.

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