d Does (2,2) lie in the overlapping region on the graph?
A
aDefining the variables: Let x be the number of hours Opal works for a photographer and let y be the number of hours she coaches a competitive soccer team.
System of inequalities:
15x+10y ≥ 90 x+y ≤ 20
B
b
C
cExample solutions: (4,12) and (8,4)
D
d No, see solution.
Practice makes perfect
a Let x be the number of hours Opal works per week for a photographer and y be the number of hours she works per week as a coach. Since Opal makes $15 per hour working for the photographer and $10 per hour coaching, we can write an expression to represent her weekly earnings.
15x+10y
Since she needs to earn at least $90, this expression must be greater than or equal to 90. With this information, we can write our first inequality.
15x+10y ≥ 90
We also know that Opal does not want to work more than 20 hours per week. Therefore, the total amount of weekly working hours must be less than or equal to 20. With this in mind, we can write a second inequality.
x+y ≤ 20
Finally, we can write a system of inequalities formed by the two inequalities we have.
15x+10y≥ 90 x+y≤ 20
b We will start by writing the equations of the boundary lines in slope-intercept form. To find the equation of a boundary line, we replace the inequality sign with an equals sign. To write an equation in slope-intercept form, we isolate the y-variable. This will help us to identify the slope and y-intercept of each line.
Inequality
Boundary Line
Slope-intercept Form
15x+10y ≥ 90
15x+10y = 90
y= - 3/2x+ 9
x+y ≤ 20
x+y = 20
y= - x+ 20
We will use the slope and y-intercept of each line to draw the graphs. Note that since the number of hours worked cannot be negative, we will only draw the lines in the first quadrant of the coordinate plane. As neither inequality is strict, both boundary lines will be solid.
To determine the region to be shaded, we will test a point in each of the inequalities. Let's start with 15x+10y≥ 90. For simplicity, we will use (0,0) as our test point. If we substitute these coordinates and obtain a true statement, we will shade the region which contains (0,0). Otherwise, we will shade the opposite region.
Since we did not obtain a true statement, we will shade the region which does not contain the point (0,0).
By following the same procedure and testing the same point for x+y≤ 20, we can tell that this time, the point does satisfy the inequality. Therefore, for this inequality we will shade the region that contains point (0,0).
The solution set for our system is the overlapping region we see above.
c To name two possible solutions, we need any two points from the overlapping area. We will plot the points in the graph and name them.
One possible solution is (4,12). This means that Opal would work for the photographer for 4 hours and coach the soccer team for 12 hours that week. Another possible solution is (8,4), which represents Opal working 8 hours for the photographer and coaching for 4 hours. Note that these are just two of many possible answers.
d To check whether (2,2) is a solution, let's plot it in our graph and see if it lies in the shaded area.
As can be seen in the graph, the point (2,2) lies outside of the shaded region on the graph. This means that it is not a solution to this system of inequalities.
