A contractor needs to fence a rectangular backyard for a client. The length of the backyard should be at least 150 ft, and the distance around should be no more than 450 ft. Which system of inequalities and graph represents the possible dimensions of the backyard?
Let's assume that the length of the backyard is represented by "x" and the width of the backyard is represented by "y".
The length of the backyard should be at least 150 ft, which means x ≥ 150.
The distance around the backyard should be no more than 450 ft, which means 2x + 2y ≤ 450.
Therefore, the system of inequalities is:
x ≥ 150
2x + 2y ≤ 450
To graph this system of inequalities, we'll plot the equations on a coordinate plane:
x = 150 represents a vertical line on the x-axis passing through the point (150,0).
2x + 2y = 450 can be written as y = (450 - 2x) / 2 = 225 - x, which represents a downward-sloping line.
Note that since x can only be greater than or equal to 150, we only need to focus on the region to the right of the vertical line passing through (150,0).
The shaded area below the line y = 225 - x and to the right of the vertical line x = 150 represents the possible dimensions of the backyard.
Here is the graph:
https://www.desmos.com/calculator/dk23uxcf22