Answer: Let's use x to represent the number of hours Kendra works as a babysitter, and y to represent the number of hours Kendra works providing tech support. Then, we can write the following system of inequalities to represent Kendra's goal:
10x + 15y >= 1000 (Kendra needs to earn at least $1000)
x + y <= 80 (Kendra can't work more than 80 hours in total)
To graph these inequalities, we can start by graphing the boundary lines for each inequality. To do this, we can rewrite each inequality as an equation and graph it as a line:
10x + 15y = 1000 (the boundary for the earnings inequality)
x + y = 80 (the boundary for the total hours inequality)
We can then shade the region that satisfies both inequalities (i.e. the region that is above the earnings line and below the total hours line). This shaded region represents all the possible combinations of hours that Kendra could work to meet her goal:
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80| .
| .
| .
| .
| .
0|___________________
0 50 80
Babysitting Hours
In the graph, the shaded region is the triangular area bounded by the x-axis, the y-axis, and the line x + y = 80. The fewest number of hours Kendra could work and still meet her goal is the point where the earnings line intersects the total hours line within the shaded region. To find this point, we can solve the system of equations:
10x + 15y = 1000
x + y = 80
One way to solve this system is to use substitution. Solving the second equation for y, we get y = 80 - x. Substituting this expression for y into the first equation, we get:
10x + 15(80 - x) = 1000
Simplifying and solving for x, we get x = 40. Substituting this value for x into the second equation, we get y = 80 - 40 = 40.
Explanation: