aSystem of Inequalities:
10 ≤ h ≤ 30 d≥ 6.5 h
B
b
C
c 25
Practice makes perfect
a Let h be the number of hours that Josefina works in a week and d be the amount of money that she earns in that week. Since we know that she works at least 10 and at most 30 hours in a week, we can write a compound inequality to represent it.
10 ≤ h ≤ 30
Since she earns $6.50 per hour and she can earn tips, if she works h hours in a week she will earn at least 6.5h that week.
d ≥ 6.5h
If we combine the inequalities, we get a system of inequalities.
10 ≤ h ≤ 30 d≥ 6.5 h
b The number of hours that Josefina works in a week is the independent variable h, and the money that she earns in a week is the dependent variable d. We want to graph the system of inequalities.
10 ≤ h ≤ 30 & (I) d≥ 6.5 h & (II)
Consider the first inequality.
10≤ h ≤ 30
The solution set of this inequality contains all points which are between vertical lines h=10 and h=30. Since the inequalities signs are non-strict, both lines are solid.
Let's graph the boundary line of the second inequality, d = 6.5h.
Since the inequality d ≥ 6.5h has a non-strict sign, the line d = 6.5h is solid. To determine which half of plane we should shade, we will use a test point.
It looks like the point ( 10, 150) would be a good test point. We will substitute this point for h and d in the given inequality and simplify. If the substitution creates a true statement, we shade the same region the test point. Otherwise, we shade the opposite region.
We will shade the region that contains the test point, which is above the boundary line.
The overlapping portion of the inequalities is the solution set of the system of inequalities.
c Suppose Josefina worked x hours in that week. With tips she should earn 6.5x+17.5 dollars for the week. It is given that she earned a total of $ 180 in total in that week. We can equate these values.
6.5x+17.5=180
To find how many hours she worked that week, we will isolate x in this equation.
