Question 1191630: Ian is working two summer jobs, making $22 per hour tutoring and making $8 per hour landscaping. In a given week, he can work a maximum of 11 total hours and must earn at least $120. Also, he must work at most 8 hours tutoring and at most 3 hours landscaping. If xx represents the number of hours tutoring and yy represents the number of hours landscaping, write and solve a system of inequalities graphically and determine one possible solution.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
x = number of hours tutoring
y = number of hours landscaping
We cannot have negative values for either x and y, so  and 
x+y = total hours worked for both jobs combined
This total cannot exceed 11 hours
Either x+y = 11 or x+y < 11
22x = amount of money made from tutoring only
8y = amount of money made from landscaping only
22x+8y = total amount made from both jobs
 since he must make at least $120
Either 22x+8y = 120 or 22x+8y > 120
System of inequalities:
To graph this system, we'll have these boundary lines
The shaded region consists of points that make all of the inequalities mentioned earlier to be true.
For , we're describing points to the right of the y axis.
For , we're describing points above the x axis.
In short, we're talking about (x,y) points in the upper right quadrant known as Q1.
For  , we'll shade below the boundary line 
For  , we'll shade above the boundary line 
With all those shading rules in mind, we get this final inequality graph:
I used GeoGebra to make this graph.
The vertex points A,B,C are at locations
A = (2.2857, 8.7143)
B = (5.4545, 0)
C = (11, 0)
Point A is the intersection of x+y=11 and 22x+8y = 120
Point B is the intersection of 22x+8y = 120 and the x axis
Point C is the intersection of x+y = 11 and the x axis
A solution point is anything in the blue shaded region.
Points on the boundary are allowed as well due to the "or equal to" as part of the inequality signs. Boundary points must be adjacent to the interior shaded region.
One such solution is (x,y) = (7,2)
Notice that x+y = 7+2 = 9 is less than 11 hours
And 22x+8y = 22*7+8*2 = 170 is larger than $120
Both conditions are met.
Therefore, for this example solution, it is possible for Ian to work 7 hours as a tutor and 2 hours as a landscaper such that he doesn't exceed the 11 hour max and he also earns $120 or more.
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