5-6 Graphing Inequalities in Two Variables
Graph each inequality.
y >x+3
y>x+3
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not greater than 3, shade the half plane that does not contain (0, 0).
y
8
y
8
8
Since 0 is greater than or equal to 8, shade the half plane that contains (0, 0).
x +y > 1
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 1
5-6 Graphing Inequalities in Two Variables
x +y > 1
Solve for y in terms of x.
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not greater than 1, shade the half plane that does not contain (0, 0).
y
x
6
y
x
6
Since 0 is not less than or equal to 6, shade the half plane that does not contain (0, 0).
eSolutions Manual - Powered by Cognero
y < 2x
4
Page 2
5-6 Graphing Inequalities in Two Variables
y
x
6
y
x
6
Since 0 is not less than or equal to 6, shade the half plane that does not contain (0, 0).
y < 2x
4
y < 2x
4
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not less than 4, shade the half plane that does not contain (0, 0).
x
y
4
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 3
5-6 Graphing Inequalities in Two Variables
x
y
4
Solve for y in terms of x.
Since 0 is less than or equal to 4, shade the half plane that contains (0, 0).
Use a graph to solve each inequality.
7x + 1 < 15
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and get 0 on a side
by itself.
Because the inequality involves <, graph x = 2 using a dashed line.
Choose (0, 0) as a test point in the original inequality.
eSolutions Manual - Powered by Cognero
.
Page 4
5-6 Graphing Inequalities in Two Variables
Use a graph to solve each inequality.
7x + 1 < 15
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and get 0 on a side
by itself.
Because the inequality involves <, graph x = 2 using a dashed line.
Choose (0, 0) as a test point in the original inequality.
.
Since 1 is less than 15, shade the half plane that contains the point (0, 0).
Notice that the x intercept of the graph is at 2. Since the half plane to the left of the x intercept is shaded, the
solution is x < 2.
3x
2
11
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and get 0 on a side
by itself.
eSolutions Manual - Powered by Cognero
Page 5
5-6 Graphing Inequalities in Two Variables
3x
2
11
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and get 0 on a side
by itself.
Because the inequality involves
raph x =
with a solid line.
Choose (0, 0) as a test point in the original inequality.
Since 2 is not greater than or equal to 11, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
3y
5
. Since the half plane to the left of the x intercept is shaded, the
.
34
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for y.
eSolutions Manual - Powered by Cognero
Page 6
5-6 Graphing Inequalities in Two Variables
3y
5
34
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for y.
Because the inequality involves
raph y = 13 using a solid line.
Choose (0, 0) as a test point in the original inequality.
.
Since 5 is less than or equal 34, shade the half plane that contains the point (0, 0).
Notice that the y intercept of the graph is at 13. Since the half plane below the y intercept is shaded, the solution is
y
4y
21 > 1
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for y.
Page 7
eSolutions Manual - Powered by Cognero
Because the inequality involves >, graph
using a dashed line.
5-6 Graphing Inequalities in Two Variables
4y
21 > 1
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for y.
Because the inequality involves >, graph
using a dashed line.
Choose (0, 0) as a test point in the original inequality.
Since 21 not is greater than 1, shade the half plane that does not contain the point (0, 0).
Notice that the y intercept of the graph is at 5
is
. Since the half plane above the y intercept is shaded, the solution
.
The surf shop has a weekly overhead of $2300.
Write an inequality to represent the number of skimboards and longboards the shop sells each week to make a
profit.
How many skimboards and longboards must the shop sell each week to make a profit?
eSolutions Manual - Powered by Cognero
Page 8
The surf shop has a weekly overhead of $2300.
Write an inequality to represent the number of skimboards and longboards the shop sells each week to make a
profit.
5-6 Graphing Inequalities in Two Variables
How many skimboards and longboards must the shop sell each week to make a profit?
a. Let x represent the skimboards and y represent the longboards. 115x + 685y
b. To graph the inequality, first solve for x.
Graph. then use the test point (0, 0) to determine the shading. The point (0, 0) results in 0
One possible solution would be to sell 1 skim board. Calculate how many longboards must be sold when just 1 skim
So, the shop would have to sell 1 skim board and 4 longboard surf boards.
Graph each inequality.
y <x 3
eSolutions Manual - Powered by Cognero
Page 9
5-6 Graphing Inequalities in Two Variables
So, the shop would have to sell 1 skim board and 4 longboard surf boards.
Graph each inequality.
y <x 3
y<x
3
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not less than 3, shade the half plane that does not contain (0, 0).
y > x + 12
y > x + 12
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not less than 12, shade the half plane that does not contain (0, 0).
y
3x
1
eSolutions Manual - Powered by Cognero
Page 10
5-6 Graphing Inequalities in Two Variables
y
3x
1
Since 0 is greater than or equal to 1, shade the half plane that contains (0, 0).
y
4x + 12
Because the inequality involves , graph the boundary using a solid line. Choose (0, 0) as a test point.
Since 0 is less than or equal to 12, shade the half plane that contains (0, 0).
6x + 3y > 12
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 11
5-6 Graphing Inequalities in Two Variables
6x + 3y > 12
Solve for y in terms of x.
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not greater than 4, shade the half plane that does not contain (0, 0).
2x + 2y < 18
Solve for y in terms of x.
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
eSolutions Manual - Powered by Cognero
Since 0 is less than 9, shade the half plane that contains (0, 0).
Page 12
5-6 Graphing Inequalities in Two Variables
2x + 2y < 18
Solve for y in terms of x.
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is less than 9, shade the half plane that contains (0, 0).
5x + y > 10
Solve for y in terms of x.
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not greater than 10, shade the half plane that does not contain (0, 0).
eSolutions Manual - Powered by Cognero
Page 13
5-6 Graphing Inequalities in Two Variables
5x + y > 10
Solve for y in terms of x.
Because the inequality involves >, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not greater than 10, shade the half plane that does not contain (0, 0).
2x + y < 3
Solve for y in terms of x.
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not less than 3, shade the half plane that does not contain (0, 0).
eSolutions Manual - Powered by Cognero
Page 14
5-6 Graphing Inequalities in Two Variables
2x + y < 3
Solve for y in terms of x.
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is not less than 3, shade the half plane that does not contain (0, 0).
2x + y
4
Solve for y in terms of x.
Because the inequality involves , graph the boundary using a solid line. Choose (0, 0) as a test point.
Since 0 is greater than or equal to 4, shade the half plane that contains (0, 0).
eSolutions Manual - Powered by Cognero
Page 15
5-6 Graphing Inequalities in Two Variables
2x + y
4
Solve for y in terms of x.
Because the inequality involves , graph the boundary using a solid line. Choose (0, 0) as a test point.
Since 0 is greater than or equal to 4, shade the half plane that contains (0, 0).
8x + y
6
Solve for y in terms of x.
Since 0 is less than or equal to 6, shade the half plane that contains (0, 0).
eSolutions Manual - Powered by Cognero
Page 16
5-6 Graphing Inequalities in Two Variables
8x + y
6
Solve for y in terms of x.
Since 0 is less than or equal to 6, shade the half plane that contains (0, 0).
10x + 2y
14
Solve for y in terms of x.
Since 0 is less than or equal to 7, shade the half plane that contains (0, 0).
eSolutions Manual - Powered by Cognero
Page 17
5-6 Graphing Inequalities in Two Variables
10x + 2y
14
Solve for y in terms of x.
Since 0 is less than or equal to 7, shade the half plane that contains (0, 0).
24x + 8y
48
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 18
5-6 Graphing Inequalities in Two Variables
24x + 8y
48
Solve for y in terms of x.
Since 0 is greater than or equal to 6, shade the half plane that contains (0, 0).
Use a graph to solve each inequality.
10x 8 < 22
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for x.
Because the inequality involves <, graph x = 3 using a dashed line.
Choose (0, 0) as a test point in the original inequality.
eSolutions Manual - Powered by Cognero
.
Page 19
5-6 Graphing Inequalities in Two Variables
Use a graph to solve each inequality.
10x 8 < 22
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for x.
Because the inequality involves <, graph x = 3 using a dashed line.
Choose (0, 0) as a test point in the original inequality.
.
Since 8 is less than 22, shade the half plane that contains the point (0, 0).
Notice that the x intercept of the graph is at 3. Since the half plane to the left of the x intercept is shaded, the
solution is x < 3.
20x
5 > 35
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for x.
eSolutions Manual - Powered by Cognero
Because the inequality involves >, graph x = 2 using a dashed line.
Choose (0, 0) as a test point in the original equation.
Page 20
5-6 Graphing Inequalities in Two Variables
20x
5 > 35
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for x.
Because the inequality involves >, graph x = 2 using a dashed line.
Choose (0, 0) as a test point in the original equation.
Since 5 is not greater than 35, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at 2. Since the half plane to the right of the x intercept is shaded, the
solution is x > 2.
4y
77
23
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for y.
eSolutions Manual - Powered by Cognero
Because the inequality involves
raph y = 25 using a solid line.
Choose (0, 0) as a test point in the original equation.
Page 21
5-6 Graphing Inequalities in Two Variables
4y
77
23
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign, and solve for y.
Because the inequality involves
raph y = 25 using a solid line.
Choose (0, 0) as a test point in the original equation.
Since 77 is not greater than or equal 23, shade the half plane that does not contain the point (0, 0).
Notice that the y intercept of the graph is at 25. Since the half plane above the y intercept is shaded, the solution is
y
5y + 8
33
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for y.
eSolutions Manual - Powered by Cognero
Page 22
5-6 Graphing Inequalities in Two Variables
5y + 8
33
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for y.
Because the inequality involves
raph y
Choose (0, 0) as a test point in the original equation.
.
Since 8 is less than or equal 33, shade the half plane that contains the point (0, 0).
Notice that the y intercept of the graph is at 5. Since the half plane under the y intercept is shaded, the solution is y
35x + 25 < 6
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
eSolutions Manual - Powered by Cognero
Page 23
Notice that the y intercept of the graph is at 5. Since the half plane under the y intercept is shaded, the solution is y
5-6 Graphing Inequalities in Two Variables
35x + 25 < 6
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves <, graph x =
using a dashed line.
Choose (0, 0) as a test point in the original inequality.
.
Since 25 is not less than 6, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
14x
. Since the half plane to the left of the x intercept is shaded, the
.
12 > 31
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
eSolutions Manual - Powered by Cognero
Page 24
Notice that the x intercept of the graph is at
. Since the half plane to the left of the x intercept is shaded, the
solution is
.
5-6 Graphing Inequalities in Two Variables
14x
12 > 31
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves >, graph x =
using a dashed line.
Choose (0, 0) as a test point in the original equation.
.
Since 12 is greater than 31, shade the half plane that contains the point (0, 0).
Notice that the x intercept of the graph is at
solution is
. Since the half plane to the right of the x intercept is shaded, the
.
Sybrina is decorating her bedroom. She has $300 to spend on paint and bed linens. A gallon of
paint costs $14, while a set of bed linens costs $60.
Write an inequality for this situation.
How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget?
a. Let x represent the number of gallons of paint and y represent the number of sets of bed linens. 14x + 60y
b. To graph the inequality, solve for x
eSolutions Manual - Powered by Cognero
Page 25
Notice that the x intercept of the graph is at
. Since the half plane to the right of the x intercept is shaded, the
5-6 Graphing
in Two Variables
solution is Inequalities
.
Sybrina is decorating her bedroom. She has $300 to spend on paint and bed linens. A gallon of
paint costs $14, while a set of bed linens costs $60.
Write an inequality for this situation.
How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget?
a. Let x represent the number of gallons of paint and y represent the number of sets of bed linens. 14x + 60y
b. To graph the inequality, solve for x
Graph. Then use the test point (0, 0) to determine the shading. The point (0, 0) results in 0
One possible solution would be to buy 5 gallons of paint Calculate how many bed linen sets must be purchased when
Use a graph to solve each inequality.
3x + 2 < 0
First,Manual
graph- Powered
the boundary,
which
eSolutions
by Cognero
x.
is the related equation. Replace the inequality sign with an equals sign and solvePage
for 26
5-6 Graphing Inequalities in Two Variables
Use a graph to solve each inequality.
3x + 2 < 0
First, graph the boundary, which is the related equation. Replace the inequality sign with an equals sign and solve for
x.
Because the inequality involves <, graph
Choose (0, 0) as a test point in the original inequality.
.
Since 2 is not less than 0, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
4x
. Since the half plane to the left of the x intercept is shaded, the
.
1>3
First, graph the boundary, which is the related equation. Replace the inequality sign with an equals sign and solve for
x.
eSolutions Manual - Powered by Cognero
Page 27
Notice that the x intercept of the graph is at
. Since the half plane to the left of the x intercept is shaded, the
solution is Inequalities
.
5-6 Graphing
in Two Variables
4x
1>3
First, graph the boundary, which is the related equation. Replace the inequality sign with an equals sign and solve for
x.
Because the inequality involves >, graph x = 1 with a dashed line.
Choose (0, 0) as a test point in the original inequality.
.
Since 1 is not greater than 3, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at 1. Since the half plane to the right of the x intercept is shaded, the
solution is x > 1.
6x
8
4
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Page 28
eSolutions Manual - Powered by Cognero
Because the inequality involves
raph x =
using a solid line.
Notice that the x intercept of the graph is at 1. Since the half plane to the right of the x intercept is shaded, the
solution is xInequalities
> 1.
5-6 Graphing
in Two Variables
6x
8
4
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves
raph x =
using a solid line.
Choose (0, 0) as a test point in the original inequality.
Since 8 is not greater than or equal to 4, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
. Since the half plane to the left of the x intercept is shaded, the
.
5x + 1 < 3
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
eSolutions Manual - Powered by Cognero
Page 29
Notice that the x intercept of the graph is at
. Since the half plane to the left of the x intercept is shaded, the
solution is
.
5-6 Graphing Inequalities in Two Variables
5x + 1 < 3
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves <, graph x =
using a dashed line.
Choose (0, 0) as a test point in the original inequality.
Since 1 is less than 3, shade the half plane that contains the point (0, 0).
Notice that the x intercept of the graph is at
solution is
. Since the half plane to the right of the x intercept is shaded, the
.
7x + 13 < 10
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
eSolutions Manual - Powered by Cognero
Page 30
Notice that the x intercept of the graph is at
. Since the half plane to the right of the x intercept is shaded, the
solution is
.
5-6 Graphing Inequalities in Two Variables
7x + 13 < 10
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves <, graph x =
using a dashed line.
Choose (0, 0) as a test point in the original inequality.
Since 13 is not less than 10, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
4x
4
. Since the half plane to the right of the x intercept is shaded, the
.
6
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
eSolutions Manual - Powered by Cognero
Page 31
Notice that the x intercept of the graph is at
. Since the half plane to the right of the x intercept is shaded, the
solution is
.
5-6 Graphing Inequalities in Two Variables
4x
4
6
Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
Because the inequality involves
raph x =
using a solid line.
Choose (0, 0) as a test point in the original inequality.
.
Since 4 is not less than or equal to 6, shade the half plane that does not contain the point (0, 0).
Notice that the x intercept of the graph is at
solution is
. Since the half plane to the left of the x intercept is shaded, the
.
The girls soccer team wants to raise $2000 to buy new goals. How many of each item must they sell to
buy the goals?
eSolutions Manual - Powered by Cognero
Page 32
solution is
.
5-6 Graphing Inequalities in Two Variables
The girls soccer team wants to raise $2000 to buy new goals. How many of each item must they sell to
buy the goals?
Write an inequality that represents this situation.
Graph this inequality.
Make a table of values that shows at least five possible solutions.
Plot the solutions from part c.
a. Let x represent the number of hot dogs and let y represent the number of sodas. x + 1.25y
2000
b.
c. Sample answer:
x
y
400
1600
200
1500
300
1400
400
1300
1000
1000
d. Sample points should be in the shaded region of the graph in part b.
Graph each inequality. Determine which of the ordered pairs are part of the solution set for each
inequality.
y 6; {(0, 4), ( 2, 7), (4, 8), ( 4, 8), (1, 6)}
y
0
6
Since 0 is not greater than or equal to 6, shade the half plane that does not contain (0, 0).
eSolutions Manual - Powered by Cognero
Page 33
200
1500
300
1400
400
1300
1000
1000
5-6 Graphing Inequalities
in Two Variables
d. Sample points should be in the shaded region of the graph in part b.
Graph each inequality. Determine which of the ordered pairs are part of the solution set for each
inequality.
y 6; {(0, 4), ( 2, 7), (4, 8), ( 4, 8), (1, 6)}
y
0
6
Since 0 is not greater than or equal to 6, shade the half plane that does not contain (0, 0).
Because ( 2, 7), (4, 8) and (1, 6) are in the shaded half plane or on the boundary line, they are part of the solution
set.
x < 4; {(2, 1), ( 3, 0), (0, 3), ( 5, 5), ( 4, 2)}
y< 4
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
0
4
eSolutions
Manual
- Powered
by Cognero
Since
0 is not
less than
4, shade
the half plane that does not contains (0, 0).
Page 34
Because ( 2, 7), (4, 8) and (1, 6) are in the shaded half plane or on the boundary line, they are part of the solution
5-6 Graphing
Inequalities in Two Variables
set.
x < 4; {(2, 1), ( 3, 0), (0, 3), ( 5, 5), ( 4, 2)}
y< 4
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
0
4
Since 0 is not less than 4, shade the half plane that does not contains (0, 0).
Because ( 5, 5) is in the shaded half plane, it is part of the solution set.
2x
3y
1; {(2, 3), (3, 1), (0, 0), (0, 1), (5, 3)}
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 35
5-6 Graphing Inequalities in Two Variables
Since 0 is greater than or equal to
, shade the half plane that contains (0, 0).
Because (2, 3), (0, 0), and (5, 3) are in the shaded half plane or on the boundary line, they are part of the solution
set.
5x + 7y
10; {( 2, 2), (1, 1), (1, 1), (2, 5), (6, 0)}
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 36
5x + 7y
10; {( 2, 2), (1, 1), (1, 1), (2, 5), (6, 0)}
5-6 Graphing Inequalities in Two Variables
Solve for y in terms of x.
Because the inequality involves
Since 0 is not greater than or equal to
, shade the half plane that does not contain (0, 0).
Because (1, 1), (2, 5), and (6, 0) are in the shaded half plane or on the boundary line, they are part of the solution
set.
3x + 5y < 10; {(3, 1), (1, 1), (0, 8), ( 2, 0), (0, 2)}
eSolutions Manual - Powered by Cognero
Solve for y in terms of x.
Page 37
Because (1,Inequalities
1), (2, 5), and
0) Variables
are in the shaded half plane or on the boundary line, they are part of the solution
5-6 Graphing
in (6,
Two
set.
3x + 5y < 10; {(3, 1), (1, 1), (0, 8), ( 2, 0), (0, 2)}
Solve for y in terms of x.
Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
Since 0 is less than 10, shade the half plane that contains (0, 0).
Because (3, 1), (1, 1), and (2, 0) are in the shaded half plane, they are part of the solution set.
2x
2y
4; {(0, 0), (0, 7), (7, 5), (5, 3), (2, 5)}
Solve for y in terms of x.
eSolutions Manual - Powered by Cognero
Page 38
5-6 Graphing Inequalities in Two Variables
Because (3, 1), (1, 1), and (2, 0) are in the shaded half plane, they are part of the solution set.
2x
2y
4; {(0, 0), (0, 7), (7, 5), (5, 3), (2, 5)}
Solve for y in terms of x.
Because the inequality involves
Since 0 is not greater than or equal to 4, shade the half plane that does not contain (0, 0).
Because (7, 5), (5, 3), and (2, 5) are in the shaded half plane or on the boundary line, they are part of the solution
set.
Mr. Jones would like to spend no more than $37.50 per week on recycling. A curbside recycling
service will remove up to 50 pounds of plastic bottles and paper products per week. They charge $0.25 per pound of
eSolutions
Manual
Powered
Cogneroof paper products.
Page 39
plastic
and-$0.75
perbypound
Write an inequality that describes the number of pounds of each product that can be included in the curbside
service.
Because (7,Inequalities
5), (5, 3), and
are in the shaded half plane or on the boundary line, they are part of the solution
5-6 Graphing
in (2,
Two5)Variables
set.
Mr. Jones would like to spend no more than $37.50 per week on recycling. A curbside recycling
service will remove up to 50 pounds of plastic bottles and paper products per week. They charge $0.25 per pound of
plastic and $0.75 per pound of paper products.
Write an inequality that describes the number of pounds of each product that can be included in the curbside
service.
Write an inequality that describes Mr. Jones weekly cost for the service if he stays within his budget.
Graph an inequality for the weekly costs for the service.
a. Let x represent the pounds of plastic and y represent the pounds of paper products. They will remove up to 50
x +y
0.25x + 0.75y
c.
Use inequalities A and B to investigate graphing compound inequalities on a
coordinate plane.
7(y + 6) 21x + 14
3y 3x 12
Solve each inequality for y.
Graph both inequalities on one graph. Shade the half plane that makes A true in red. Shade the
half plane that makes B true in blue.
What does the overlapping region represent?
a. For equation A:
For equation B:
b.
eSolutions Manual - Powered by Cognero
Page 40
5-6 Graphing Inequalities in Two Variables
Use inequalities A and B to investigate graphing compound inequalities on a
coordinate plane.
7(y + 6) 21x + 14
3y 3x 12
Solve each inequality for y.
Graph both inequalities on one graph. Shade the half plane that makes A true in red. Shade the
half plane that makes B true in blue.
What does the overlapping region represent?
a. For equation A:
For equation B:
b.
c. The overlapping region represents the solutions that make both A and B true.
Reiko and Kristin are solving
by graphing. Is either of them correct? Explain your
reasoning.
Kristin used a test point located on the line and shaded the incorrect half-plane. Reiko is correct.
CHALLENGE Write a linear inequality for which ( 1, 2), (1, 1), and (3, 4) are solutions but (0, 1) is not.
eSolutions Manual - Powered by Cognero
Page 41
We want to find a linear inequality for which three points are solutions and one point is not a solution. The best way
5-6 Graphing Inequalities in Two Variables
Kristin used a test point located on the line and shaded the incorrect half-plane. Reiko is correct.
CHALLENGE Write a linear inequality for which ( 1, 2), (1, 1), and (3, 4) are solutions but (0, 1) is not.
We want to find a linear inequality for which three points are solutions and one point is not a solution. The best way
to find this is to plot the points on the coordinate plane and visualize the line that would be satisfactory.
Plot all of the points and label (1,1) with an X to separate it from the others.
From the graph, it looks like the three points that are included in the solution almost form a straight line. It looks like a
good line to use would be the one that includes ( 1, 2) and (0, 1). It also looks like point X would be above this line
and the other point would be below it. Therefore, the linear inequality should be close to this line and the shading
should be below it.
Using points ( 1, 2) and (0, 1), you can determine the equation for the line to be
are a part of the solution and we are shading below the line, so the inequality is
. The points on the line
.
Explain why a point on the boundary should not be used as a test point.
The test point is used to determine which half plane is the solution set for the inequality. A test point on the boundary
does not show which half plane contains the points that make the inequality true.
Write a two-variable inequality with a restricted domain and range to represent a real-world
situation. Give the domain and range, and explain why they are restricted.
Sample answer: The inequality y > 10x + 45 represents the cost of a monthly smartphone data plan with a flat rate
of $45 for the first 2 GB of data used, plus $10 per each additional GB of data used. Both the domain and range are
nonnegative real numbers because the GB used, and the total cost cannot be negative.
Summarize the steps to graph an inequality in two variables.
eSolutions Manual - Powered by Cognero
Page 42
Sample answer: First solve the inequality for y. Then change the inequality sign to an equal sign and graph the
boundary. If < or > is used, the boundary is not included in the graph and the line is dashed. Otherwise, the boundary
Sample answer: The inequality y > 10x + 45 represents the cost of a monthly smartphone data plan with a flat rate
of $45 for the first 2 GB of data used, plus $10 per each additional GB of data used. Both the domain and range are
nonnegative
real numbers
the GB used, and the total cost cannot be negative.
5-6 Graphing
Inequalities
in because
Two Variables
Summarize the steps to graph an inequality in two variables.
Sample answer: First solve the inequality for y. Then change the inequality sign to an equal sign and graph the
boundary. If < or > is used, the boundary is not included in the graph and the line is dashed. Otherwise, the boundary
is included and the line is solid. Then choose a test point not on the boundary.
Substitute the coordinates of the test point into the original inequality. If the result makes the inequality true, then
shade the half-plane that includes the test point. If the result makes the inequality false, shade the half-plane that
does not include the test point. Lastly, check your solution by choosing a test point that is in the half-plane that is not
shaded. This second test point should make the inequality false if the solution is correct.
What is the domain of this function?
{x | 0 x
{x | 0 x
{y | 0 y
{y | 0 y
3}
9}
9}
3}
The domain of a function is the x values, which means that you can delete choices C and D. This graph shows
values from 0 to 9, so the correct choice is B.
An arboretum will close for the winter when all of the trees have lost their leaves. The
table shows the number of trees each day that still have leaves.
Write an equation that represents the number of trees with leaves y after d days.
Find the y-intercept. What does it mean in the context of this problem?
After how many days will the arboretum close? Explain how you got your answer.
a. Choose two points (5, 325) and (20, 130). Find the slope of the line containing the given points.
eSolutions Manual - Powered by Cognero
Use the slope and either of the two points to find the y intercept.
Page 43
Find the y-intercept. What does it mean in the context of this problem?
After how many days will the arboretum close? Explain how you got your answer.
5-6 Graphing Inequalities in Two Variables
a. Choose two points (5, 325) and (20, 130). Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
Write the equation in slope intercept form.
b. The y intercept is 390. This represents the number of trees in the arboretum before any have lost their leaves.
c. To determine when the arboretum will close, let y = 0.
The arboretum will close after 30 days.
Which inequality best represents the statement below?
A jar contains 832 gumballs. Ebony s guess was within 46 pieces.
|g 832| 46
|g + 832| 46
|g 832| 46
|g + 832| 46
The guess was within 46, means less than or equal to 46, which eliminates choices H and J. For |g
g+
g -786. The number of gumballs cannot be negative, so G is eliminated.
The correct choice is F.
eSolutions Manual - Powered
by Cognero
If the
rectangular
3
prism has a volume of 10,080 cm , what is the value of x?
Page 44
The guess was within 46, means less than or equal to 46, which eliminates choices H and J. For |g
g+
g -786. The number of gumballs cannot be negative, so G is eliminated.
5-6 Graphing Inequalities in Two Variables
The correct choice is F.
3
If the rectangular prism has a volume of 10,080 cm , what is the value of x?
12
14
16
18
Volume of a rectangular prism is equal to length times width times height.
So, the correct choice is D.
Solve each open sentence.
|y 2| > 4
Case 1:
Case 2:
So the solution set is {y | y > 6 or y < 2}.
|t
6|
5
Case 1:
Case 2:
eSolutions Manual - Powered by Cognero
Page 45
5-6 Graphing
Inequalities
So the solution
set is {y |iny Two
> 6 orVariables
y < 2}.
|t
6|
5
Case 1:
Case 2:
So the solution set is {t
t
|3 + d| < 4
This problem has an absolute value equal to a negative number. That means that the distance between 3 and d is
equal to 4, but distances cannot be negative, so there is no solution.
Solve each compound inequality.
4c 4 < 8c 16 < 6c 6
So, the solution set is {c | 3 < c < 5}.
eSolutions Manual - Powered by Cognero
Page 46
5-6 Graphing Inequalities in Two Variables
So, the solution set is {c | 3 < c < 5}.
So, the solution set is {p | 4 < p < 10}.
0.5n
7 or 2.5n + 2
9
Notice the two inequalities overlap so that every point is a solution. So, the solution set is {n | n is any real number}.
Write an equation of the line that passes through each pair of points.
(1, 3) and (2, 5)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
eSolutions Manual - Powered by Cognero
Page 47
5-6 Graphing Inequalities in Two Variables
Notice the two inequalities overlap so that every point is a solution. So, the solution set is {n | n is any real number}.
Write an equation of the line that passes through each pair of points.
(1, 3) and (2, 5)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
Write the equation in slope intercept form.
( 2, 4) and ( 7, 3)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
eSolutions Manual - Powered by Cognero
Page 48
Write the equation in slope intercept form.
5-6 Graphing Inequalities in Two Variables
( 2, 4) and ( 7, 3)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
Write the equation in slope intercept form.
( 6, 8) and ( 8, 5)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
eSolutions Manual - Powered by Cognero
Page 49
Write the equation in slope intercept form.
5-6 Graphing Inequalities in Two Variables
( 6, 8) and ( 8, 5)
Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
Write the equation in slope intercept form.
The table shows the maximum heart rate to maintain during aerobic activities. Write an equation in
function notation for the relation. Determine what would be the maximum heart rate to maintain in aerobic training
for an 80-year-old.
Choose two points: (20, 175) and (70, 130). Find the slope of the line containing the given points.
eSolutions Manual - Powered by Cognero
Use the slope and either of the two points to find the y intercept.
Page 50
Write the equation in slope intercept form.
5-6 Graphing Inequalities in Two Variables
The table shows the maximum heart rate to maintain during aerobic activities. Write an equation in
function notation for the relation. Determine what would be the maximum heart rate to maintain in aerobic training
for an 80-year-old.
Choose two points: (20, 175) and (70, 130). Find the slope of the line containing the given points.
Use the slope and either of the two points to find the y intercept.
Write the equation in function notation.
To determine what the maximum heart rate is for an 80 year old, substitute 80 in for x.
The maximum heart rate for an 80-year-old is 121 beats/min.
The formula
is used to find keyboarding speeds. In the formula, s represents the speed in
words per minute, w the number of words typed, r
for r.
eSolutions Manual - Powered by Cognero
m the number of minutes typed. Solve
Page 51
5-6 Graphing Inequalities in Two Variables
The maximum heart rate for an 80-year-old is 121 beats/min.
The formula
is used to find keyboarding speeds. In the formula, s represents the speed in
words per minute, w the number of words typed, r
for r.
eSolutions Manual - Powered by Cognero
m the number of minutes typed. Solve
Page 52