Let xx represent the number of hours babysitting and yy represent the number of hours clearing tables.
According to the problem, the following conditions must be satisfied:
1) $7xx + $10yy ≥ $90 (minimum amount of money earned)
2) xx + yy ≤ 12 (maximum total number of hours worked)
In order to graphically represent these inequalities, we first need to solve each inequality for yy.
1) $7xx + $10yy ≥ $90
Solving for yy: $10yy ≥ $90 - $7xx
yy ≥ ($90 - $7xx)/$10
2) xx + yy ≤ 12
Solving for yy: yy ≤ 12 - xx
Now we can graph each inequality separately.
For the first inequality, we have:
yy ≥ ($90 - $7xx)/$10
Let's graph this inequality:
First, plot the line y = ($90 - $7xx)/$10:
- When xx = 0, yy = ($90 - $7(0))/$10 = $90/$10 = $9 (point: (0, 9))
- When xx = 10, yy = ($90 - $7(10))/$10 = $20/$10 = $2 (point: (10, 2))
Plotting these two points on a graph and drawing a line passing through them, we get:
```
y
|\
| \
| \
| \
| \
| \
| \
------------------ x
```
The line should be solid because yy is greater than or equal to ($90 - $7xx)/$10.
For the second inequality, we have:
yy ≤ 12 - xx
Let's graph this inequality:
First, plot the line y = 12 - xx:
- When xx = 0, yy = 12 - 0 = 12 (point: (0, 12))
- When xx = 10, yy = 12 - 10 = 2 (point: (10, 2))
Plotting these two points on a graph and drawing a line passing through them, we get:
```
y
| /
| /
| /
| /
| /
|/
------------------ x
```
The line should be dashed because yy is less than or equal to 12 - xx.
Now we can shade the region that satisfies both inequalities.
```
y
|\
| \
| \ /
| \ /
| \/
|
|
------------------ x
```
The shaded region represents the possible solutions.
One possible solution within this region could be: xx = 6 and yy = 6. This means Mason works 6 hours babysitting and 6 hours clearing tables.
Mason is working two summer jobs, making $7 per hour babysitting and making $10 per hour clearing tables. In a given week, he can work at most 12 total hours and must earn no less than $90. If xx represents the number of hours babysitting and yy represents the number of hours clearing tables, write and solve a system of inequalities graphically and determine one possible solution.
1 answer
1 answer