a Write three inequalities, one that represents the number of surfperch you are allowed to catch, one that represents the number of rockfish, and one that represents the total number of fish. Then, graph each inequality using the boundary line and a test point.
aSystem of Inequalities: x ≤ 15 & (I) y ≤ 10 & (II) x+y ≤20 & (III)
Graph:
B
b Yes, see solution.
Practice makes perfect
a We will have three inequalities. The first will represent the number of surfperch allowed to catch, the second will represent the number of rockfish, and the third will represent the total number of fishes. Let x be the number of surfperch and y be the number of rockfish allowed to be caught in one day. We can now form our inequalities.
Inequality (I): We are told that we can catch no more than 15 surfperch per day. The phrase "no more than" is represented by ≤. With this information, we can write the following inequality.
x ≤ 15
Inequality (II): We also know that we can catch no more than 10 rockperch per day. Let's write the following inequality.
y ≤ 10
Inequality (III): Lastly, we can catch no more than 20 total fish per day. Therefore, we can write the following inequality.
x+y ≤ 20
The three inequalities we have written so far form a system of linear inequalities.
x ≤ 15 & (I) y ≤ 10 & (II) x+y≤20 & (III)
Graphing the System
To graph a system of inequalities, graph each inequality separately. The solution to the system is the intersection of the individual solution sets.
Graphing Inequality (I)
Before graphing the inequality, we need the boundary line. By replacing the inequality sign with an equals sign, we get the boundary line.
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Inequality & & Boundary Line
x ≤ 15 & & x = 15
Note that the boundary line is a vertical line. Moreover, since the inequality is not strict, the line is solid. Since x is less than or equal to 15, we will shade the part of the plane which is to the left of the line. We will only consider the first quadrant, since it is impossible for the number of surfperch to be negative.
Graphing Inequality (II)
To graph Inequality (II), we will obtain the boundary line by changing the inequality sign with an equals sign.
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Inequality & & Boundary Line
y ≤ 10 & & y = 10
Note that the boundary line is a horizontal line. Moreover, since the inequality is not strict, the line is solid. Since y is less than or equal to 10, we will shade the part of the plane which is below the boundary line.
Graphing Inequality (III)
To graph Inequality (III), 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.
Solution to the system
The solution set is the area where all of the inequalities in the system overlap.
b If it is possible to catch 11 surfperch and 9 rockfish in one day, and satisfy the restraints of both inequalities, the point (11,9) will be a solution to the system. One way we can determine if this is true is by checking if (11,9) lies in the shaded region of the graph.
Notice that the point lies on the boundary line. Since the line is solid and not dashed, the point is considered a solution to the system. Therefore, it is possible to catch 11 surfperch and 9 rockfish in one day.
