Question

Morgan is working two summer jobs, making $20 per hour tutoring and making $12 per hour clearing tables. In a given week, she can work no more than 16 total hours and must earn at least $240. If x represents the number of hours tutoring and y represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.

Answer 1

Explanation:

Answer 2

• x + y ≤ 16
• 20x +12y ≥ 240
• a graph is attached
• possible solutions: (x, y) = (6, 10), (12, 0), (16, 0), (10, 5)

Explanation:

You want a set of inequalities, their graph, and a possible solution that represents the constraints on Morgan's working two jobs. Morgan earns $20 for each hour (x) spent tutoring, and $12 for each hour (y) clearing tables. Her total hours must not exceed 16, and her total earnings must be at least $240.

Inequalities

The limit on total hours can be expressed as ...

x + y ≤ 16

The requirement on earnings can be expressed as ...

20x +12y ≥ 240

Graph

A graph of the inequalities is attached. The values given lend themselves to graphing using the intercepts of each boundary line. These are found by setting one variable to zero and solving for the other.

x + y = 16 . . . . x-intercept = 16; y-intercept = 16

20x +12y = 240 . . . . x-intercept = 240/20 = 12; y-intercept = 240/12 = 20

The first inequality is shaded below the (solid) line, and the second inequality is shaded above the line.

Possible solution

Each of the vertices marked on the graph is a possible solution, as are any points in the doubly-shaded area: (10, 5) for example.

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Additional comment

The second inequality can be "reduced" by removing a factor of 4 from each number:

5x +3y ≥ 60

This inequality has the same graph as the original. It might be said to be in "standard form" (with mutually prime coefficients), but the relationship between the coefficients and the values in the problem is lost.