Begin by writing a system of inequalities for the situation. He cannot work more than 25 hours per week. Thus, he cannot work more than 25* 8=200 hours in 8-week period.
Practice makes perfect
We will first write a system of inequalities. Let x be the number of hours working at the Pizza Palace and y be the number of hours working at a car wash. We have been told that he cannot work more than 25 hours per week. Thus, he cannot work more than 25* 8= 200 hours in 8-week period.
Number of Hours
Total Earning
Verbal Expression
Algebraic Expression
Verbal Expression
Algebraic Expression
Number of hours working at the Pizza Palace
x
Earning from working at the Pizza Palace x hours ($)
9 x
Number of hours working at a car wash
y
Earning from working at a car wash y hours ($)
12 y
Total number of working hours is less than or equal to 200.
x+ y≤ 200
Total earning from working is greater than or equal to $925.
9 x+ 12 y≥ 925
Now we have two inequalities to write a system
x+y≤ 200 & (I) 9x+12y ≥ 925 & (II)
Let's begin by graphing Inequality I. We will first find the boundary line by replacing the inequality sign with the equals sign.
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Inequality I &Boundary Line I
x+y ≤ 200 & x+y = 200
We will graph the boundary line by finding its intercepts. We will substitute y for 0 for the x-intercept and x for 0 for the y-intercept.
x+y=200
Operation
x-intercept
y-intercept
Substitution
x+ 0=200
0+y=200
Calculation
x=200
y=200
Point
(200,0)
(0,200)
Now that we know the intercepts of the boundary line, we can plot the intercepts and draw a line that passes through the them. Notice that the number of working hours cannot be negative, so the line will be bound by the axes. It will also be solid because the inequality is non-strict.
Next, we will test an arbitrary point to decide which region we should shade. Let's test the point (0,0).
The point satisfied the inequality, so we will shade the region that contains the point.
To graph the second inequality, we will think the same way.
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Inequality II &Boundary Line II
9x+12y ≥ 925 & 9x+12y = 925
Let's begin by finding the intercepts of the boundary line. We will substitute y for 0 for the x-intercept and x for 0 for the y-intercept.
9x+12y=925
Operation
x-intercept
y-intercept
Substitution
9x+12( 0)=925
9( 0)+12y=925
Calculation
x=102.8
y=77.1
Point
(102.8,0)
(0,77.1)
Now the boundary line can be graphed.
We will again test the point (0,0) to complete the graph.
