Question

Mav is working two summer jobs, making

$15 per hour lifeguarding and making $10
per hour landscaping. In a given week, she
can work at most 12 total hours and must
earn no less than $140. If x represents the
number of hours lifeguarding and y
represents the number of hours
landscaping, write and solve a system of
inequalities graphically and determine
one possible solution.
Inequality 1: y ≥û
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
y
Inequality 2: y ≥û
0 1 2 3
4
5
switch shade
plot
6
switch shade
plot
7
8 9 10 11 12 13 14 15 16 17 18 19 20
Xx

Answer 1

• y ≤ 12 -x
• y ≥ 14 -1.5x
• (x, y) = (4, 8) — 4 hours lifeguarding, 8 hours landscaping

Explanation:

You want inequalities, their graph, and a possible solution satisfying Mav's requirement for at most 12 total hours spent lifeguarding at $15 per hour and landscaping at $10 per hour, with an income of at least $140.

Setup

The required relations can be written ...

x + y ≤ 12 . . . . . . . total hours

15x +10y ≥ 140 . . . . total earnings

Inequalities

The inequalities need to be solved for y to match the answer format requirements.

inequality 1: y ≤ 12 -x . . . . . . . . . . subtract x

inequality 2: y ≥ 14 -1.5x . . . . . . divide by 10, subtract 1.5x

Solution

The attached graph shows a couple of possible solutions. The solution space is the doubly-shaded area bounded by vertices ...

(4, 8), (12, 0), and (9 1/3, 0)

Mav will make the most money working 12 hours lifeguarding (x, y) = (12, 0).

She can just meet her income requirements by working 8 hours lifeguarding and 2 hours landscaping (x, y) = (8, 2).

One possible solution is 4 hours lifeguarding and 8 hours landscaping.