What is the maximum number of waste products that will be removed by the service?
B
What is the total cost of recycling for a variable number of items? Can it exceed his budget?
C
Identify the constraints and draw the graph for the inequalities. Check which part of the graphs satisfies the inequalities. Then find the common region.
A
x+y ≤ 50
B
0.25x+ 0.75y ≤ 37.50
C
Practice makes perfect
Let x represent the amount of plastic bottles and y represent the amount of paper products Mr. Jones wants to be removed. The recycling service will not remove more than 50 pounds of waste per week. This constraint can be expressed as an inequality.
x+y ≤ 50
Mr. Jones can spend at most $37.50 per week on recycling. We can write an expression for the weekly total cost of recycling.
0.25 * x+ 0.75 * y
This amount should not be more than $37.50, and this constraint can also be expressed as an inequality.
0.25x+ 0.75y ≤ 37.50
We will proceed in steps to graph the inequalities.
First Inequality
Step 1
First we will graph the boundary, which is the related equation. Replace the inequality sign with an equals sign, and solve for y in terms of x.
We will graph the boundary line first.
y=- 4/3x+50
Since the number of items cannot be a negative number, the domain and range must contain non-negative numbers. Since the inequality is not strict, we will draw a bold boundary line.
Step 2
Let's choose (0,0) as the test point, as it is the most convenient one for the substitution.
We will graph the boundary line first.
y = - x + 50
Since the number of items cannot be a negative number, the domain and range must contain non-negative numbers. Since the inequality is not strict, we will draw a bold boundary line.
Step 2
Let's choose (0,0) as the test point, as it is the most convenient one for the substitution.
Since this result is true, we will shade the region that contains the point (0,0).
Now let's draw these two inequalities in the same coordinate system.
The solution set, which is the recycling cost not exceeding Mr.Jones' budget and not exceeding 50 pounds of waste, is the region between the x-axis, the y-axis, and the boundary line y=- 43x+50, since it is shaded twice.
