6-6 Systems of Inequalities
Solve each system of inequalities by graphing.
12. y ≤ x + 10
y > 6x + 2
SOLUTION: Graph each inequality. The graph of y ≤ x + 10 is solid and is included in the graph of the solution. The graph of y > 6x + 2 is dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≤ x + 10 and y > 6x + 2. Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below.
eSolutions Manual - Powered by Cognero
Page 1
6-6 Systems of Inequalities
The solution region is shaded in the graph below.
15. 3x − y > −5
5x − y < 9
SOLUTION: Graph each inequality. The graph of 3x − y > −5 is dashed and is not included in the graph of the solution. The graph
of 5x − y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of
ordered pairs in the intersection of the graphs of 3x − y > −5 and 5x − y < 9. The solution region is the darkest
shaded area in the graph below.
18. y ≥ 3x − 5
3x − y > −4
SOLUTION: Graph each inequality. The graph of y ≥ 3x − 5 is solid and is included in the graph of the solution. The graph of 3x − y > −4 is dashed and is not included in the graph of the solution. eSolutions Manual - Powered by Cognero
Page 2
6-6 Systems of Inequalities
18. y ≥ 3x − 5
3x − y > −4
SOLUTION: Graph each inequality. The graph of y ≥ 3x − 5 is solid and is included in the graph of the solution. The graph of 3x − y > −4 is dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≥ 3x − 5 and 3x − y > −4.
Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below.
21. y > 2x − 3
2x − y ≥ 1
SOLUTION: eSolutions
Manual - Powered by Cognero
Graph each inequality. The graph of y > 2x − 3 is dashed and is not included in the graph of the solution. Page 3
6-6 Systems of Inequalities
21. y > 2x − 3
2x − y ≥ 1
SOLUTION: Graph each inequality. The graph of y > 2x − 3 is dashed and is not included in the graph of the solution. The graph of 2x − y ≥ 1 is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x − 3 and 2x − y ≥ 1. Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below.
eSolutions
y < −2- Powered by Cognero
24. 4x +Manual
y > −4x
Page 4
6-6 Systems of Inequalities
24. 4x + y < −2
y > −4x
SOLUTION: Graph each inequality. The graph of 4x + y < −2 is dashed and is not included in the graph of the solution. The graph
of y > −4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of
ordered pairs in the intersection of the graphs of 4x + y < −2 and y > −4x. This region is shaded in the graph below.
Notice that the graphs do not intersect, so there is no solution.
Solve each system of inequalities by graphing.
27. x + y ≥ 1
x + y ≤ 2
SOLUTION: Graph each inequality. The graph of x + y ≥ 1 is solid and is included in the graph of the solution. The graph of x + y ≤ 2 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ≥ 1 and x + y ≤ 2.
eSolutions Manual - Powered by Cognero
Page 5
6-6 Systems of Inequalities
Solve each system of inequalities by graphing.
27. x + y ≥ 1
x + y ≤ 2
SOLUTION: Graph each inequality. The graph of x + y ≥ 1 is solid and is included in the graph of the solution. The graph of x + y ≤ 2 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ≥ 1 and x + y ≤ 2.
This region is shaded in the graph below.
30. y < 4x + 13
4x − y ≥ 1
eSolutions Manual - Powered by Cognero
SOLUTION: Graph each inequality. The graph of y < 4x + 13 is dashed and is not included in the graph of the solution. Page 6
6-6 Systems of Inequalities
30. y < 4x + 13
4x − y ≥ 1
SOLUTION: Graph each inequality. The graph of y < 4x + 13 is dashed and is not included in the graph of the solution. The graph of 4x − y ≥ 1 is solid. Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x − y ≥ 1, the line 4x − y
≥ 1 is included in the solution. Overlay the graphs and locate the green region. This is the intersection. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x − y ≥ 1. The solution region is shaded in the graph below.
33. eSolutions Manual - Powered by Cognero
SOLUTION: Page 7
6-6 Systems of Inequalities
33. SOLUTION: Graph each inequality. The graph of
is dashed and is not included in the graph of the solution. The graph
of
is solid and is included in the graph of the solution. The solution of the system is the set of ordered
pairs in the intersection of the graphs of
and y ≤ 9x + 2. The solution region is the darkest shaded area
in the graph below.
36. CLASS PROJECT An economics class formed a company to sell school supplies. They would like to sell at least
20 notebooks and 50 pens per week, with a goal of earning at least $60 per week.
a. Define the variables, and write a system of inequalities to represent this situation.
b. Graph the system.
c. Name one possible solution.
SOLUTION: a. Let n = the number of notebooks and p = the number of pens; n ≥ 20, p ≥ 50, $2.50n + $1.25p ≥ 60.
b.
eSolutions Manual - Powered by Cognero
Page 8
c. Name one possible solution.
SOLUTION: a. Let n of
= the
number of notebooks and p = the number of pens; n ≥ 20, p ≥ 50, $2.50n + $1.25p ≥ 60.
6-6 Systems
Inequalities
b.
c. Sample answer: 25 notebooks and 100 pens
39. REASONING State whether the following statement is sometimes, always, or never true. Explain your answer
with an example or counterexample.
Systems of inequalities with parallel boundaries have no solutions.
SOLUTION: The following system of equations have parallel boundaries and have no solution since the shaded regions do not
overlap.
2x – y > 3 and 2x – y < –1
However, the system given by y ≤ 3 and y ≤ –2 also have parallel boundaries but the solution will contain an infinite
eSolutions Manual - Powered by Cognero
number of points as one of the shaded regions completely overlaps the other. Page 9
6-6 Systems of Inequalities
39. REASONING State whether the following statement is sometimes, always, or never true. Explain your answer
with an example or counterexample.
Systems of inequalities with parallel boundaries have no solutions.
SOLUTION: The following system of equations have parallel boundaries and have no solution since the shaded regions do not
overlap.
2x – y > 3 and 2x – y < –1
However, the system given by y ≤ 3 and y ≤ –2 also have parallel boundaries but the solution will contain an infinite
number of points as one of the shaded regions completely overlaps the other. The solution would be all the points such that y ≤ –2.
Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the
solution could be one of the regions determined by the parallel boundaries.
42. CCSS PRECISION Graph the system of inequalities. Estimate the area of the solution.
y ≥ 1
y ≤ x + 4
y ≤ −x + 4
SOLUTION: Manual - Powered by Cognero
eSolutions
Page 10
To estimate the area, count up the total number of squares and half–squares. There are 6 whole squares and 6 half
squares. 6 + 6(0.5) = 9. So, the area is
.
The solution would be all the points such that y ≤ –2.
6-6 Systems
ofitInequalities
Therefore,
is sometimes true that systems with parallel boundaries have no solution. It is also possible that the
solution could be one of the regions determined by the parallel boundaries.
42. CCSS PRECISION Graph the system of inequalities. Estimate the area of the solution.
y ≥ 1
y ≤ x + 4
y ≤ −x + 4
SOLUTION: To estimate the area, count up the total number of squares and half–squares. There are 6 whole squares and 6 half
squares. 6 + 6(0.5) = 9. So, the area is
.
43. Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends that
for a person her age, her heart rate should stay within the following range as she exercises.
• It should be higher than 102 beats per minute.
• It should not exceed 174 beats per minute.
WRITING IN MATH Explain what each colored region of the graph represents. Explain how shading in various
colors can help to clearly show the solution set of a system of inequalities.
SOLUTION: Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region
represents the beats per minute above the target heart rate. The green region represents the beats per minute within
the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of
inequalities.
eSolutions Manual - Powered by Cognero
Page 11