Let x be the number of hours we work at the grocery store in a week. Since we are required to work there at least 8 hours per week, we know that x must be greater than or equal to 8. We can write our first inequality using this information.
x ≥ 8
Let y be the number of hours we teach music lessons in a week. We do not want to work more than 20 hours per week. Therefore, the total number of hours, which is the addition of x and y, must be less than or equal to 20.
x+y ≤ 20
We get paid $10 per hour worked at the grocery store and our earnings obtained from this job can be expressed as 10x. Moreover, we charge $15 per hour for teaching music lessons. Our earnings from teaching lessons can be expressed as 15y. We are told that our total earnings must be greater than or equal to 120.
10x+15y ≥ 120
The three inequalities we have written so far form a system of linear inequalities.
x≥ 8 & (I) x+y≤ 20 & (II) 10x+15y ≥ 120 & (III)
Graphing the System
To graph the system, we will consider each inequality separately. Let's start with the easiest, which is Inequality (I). To obtain the boundary line we replace the inequality sign with an equals sign.
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Inequality & & Boundary Line
x ≥ 8 & & x = 8
Note that the boundary line is a vertical line. Moreover, since the inequality is not strict, the line is solid. Since x is greater than or equal to 8, we will shade the half-plane which is to the right of the line. We will only consider the first quadrant, since it is impossible for the number of hours to be negative.
Now, consider Inequality (II). Again, we will obtain the boundary line by replacing the inequality sign with an equals sign.
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Inequality & & Boundary Line
x+y ≤ 20 & & x+y = 20
Let's rewrite the line in slope-intercept form.
The slope of this line is - 1 and the y-intercept 20. Let's use this information to draw its graph. Since the inequality is not strict, the line is solid.
To decide the region we should shade, we will test a point. If substituting the coordinates of the point in the inequality produces a true statement, we will shade the region which contains it. Otherwise, we will shade the opposite region. For simplicity, we will test the point (0,0).
Since we obtained a true statement, we will shade the region below the boundary line.
We will follow the same procedure to obtain the information we need to graph Inequality (III).
Inequality
10x+15y≥ 120
Boundary Line
10x+15y=120
Slope-intercept Form
y=- 2/3x+8
Solid or Dashed?
Solid
Test Point
(0,0)
True or False Statement?
False *
Let's graph the third inequality using the above information.
The solution set is the area where all of the inequalities in the system overlap.
b Let's use our graph to identify a solution to the system.
We see above that the point (10,8) is a solution to the system. Recall that x is the number of hours we work at the grocery store. Also, y is the number of hours we teach music. Therefore, the solution (10,8) means that we can work 10 hours at the grocery store and teach music lessons for 8 hours.
c Let's use our graph to determine whether we can work 8 hours at the grocery store and teach 1 hour of music lessons.
We see that the point (8,1) is not in the shaded area. Therefore, it is not part of the solution. This means we cannot work 8 hours at the grocery store, teach 1 hour of music lessons, and still make enough money.
