a There are 3 inequalities. Two of them will involve one variable, and the remaining one will involve two variables.
B
b Where do the inequalities overlap?
C
c Where should we search for a solution to this system?
A
aExample variables: n is the number of notebooks sold per week and p the number of pens sold per week.
System of inequalities:
n ≥ 20 p ≥ 50 2.5n+1.25p ≥ 60
B
b
C
cExample Solution: (40,100)
Practice makes perfect
a Let n be the number of notebooks sold per week and let p be the number of pens sold per week. The students would like to sell at least 20 notebooks and 50 pens per week. Therefore, n must be greater than or equal to 20, and p must be greater than or equal to 50.
n ≥ 20 and p ≥ 50
The price of one notebook is $2.50. Therefore, the expression 2.5n represents the amount earned by selling notebooks. Similarly, since the price of one pen is $1.25, the expression 1.25p represents the amount earned by selling pens. We are told the goal is earning at least $60.
2.5n+1.25p ≥ 60
We can combine the three inequalities we have written to form a system of inequalities.
n ≥ 20 & (I) p ≥ 50 & (II) 2.5n+1.25p≥ 60 & (III)
b Let the horizontal axis be the n-axis and the vertical axis be the p-axis. Let's graph the inequalities one at a time.
Inequality (I)
To obtain the boundary line we replace the inequality sign with an equals sign.
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Inequality & & Boundary Line
n ≥ 20 & & n = 20
The line n=20 is a vertical line which passes through (20,0). Since the inequality is not strict the line will be solid. The inequality states that n is greater than or equal to 20. Therefore, we will shade the half-plane to the right of the line.
Inequality (II)
The boundary line related to the second inequality is p=50. This is a horizontal line whose y-intercept is 50. Since p is greater than or equal to 50, we will shade the region above the line. It will be solid because the inequality is not strict.
Now we consider the boundary line.
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Inequality & & Boundary Line
p ≥ - 2n+48 & & p = - 2n+48
Note that the line is written in slope-intercept form. To graph it we will plot its y-intercept 48 and use the slope - 2 to find another point on the line. Then we will connect these points with a straight edge. The line will be solid, because the inequality is not strict.
To determine the half-plane we should shade we will test a point. If substituting the coordinates of this point into the inequality produces a true statement, we will shade the region that contains the point. Otherwise, we will shade the opposite region. For simplicity, we will test the point (0,0).
