5-6 Graphing Inequalities in Two Variables
Graph each inequality.
12. y < x − 3
SOLUTION: 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).
15. y ≤ −4x + 12
SOLUTION: 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).
18. 5x + y > 10
SOLUTION: Solve for y in terms of x.
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5-6 Graphing Inequalities in Two Variables
18. 5x + y > 10
SOLUTION: 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).
21. 8x + y ≤ 6
SOLUTION: 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 less than or equal to 6, shade the half–plane that contains (0, 0).
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5-6 Graphing Inequalities in Two Variables
21. 8x + y ≤ 6
SOLUTION: 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 less than or equal to 6, shade the half–plane that contains (0, 0).
Use a graph to solve each inequality.
24. 10x − 8 < 22
SOLUTION: 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.
.
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Since –8 is less than 22, shade the half–plane that contains the point (0, 0).
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5-6 Graphing Inequalities in Two Variables
Use a graph to solve each inequality.
24. 10x − 8 < 22
SOLUTION: 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.
27. 5y + 8 ≤ 33
SOLUTION: Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for y.
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Because the inequality involves ≤, graph y = 5 using a solid line.
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5-6 Graphing Inequalities in Two Variables
27. 5y + 8 ≤ 33
SOLUTION: 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 y = 5 using a solid line.
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
≤ 5.
30. DECORATING 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.
a. Write an inequality for this situation.
b. How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget?
SOLUTION: a. Let x represent the number of gallons of paint and y represent the number of sets of bed linens. 14x + 60y ≤ 300
b. To graph the inequality, solve for x first. eSolutions Manual - Powered by Cognero
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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.
5-6 Graphing
Inequalities in Two Variables
30. DECORATING 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.
a. Write an inequality for this situation.
b. How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget?
SOLUTION: a. Let x represent the number of gallons of paint and y represent the number of sets of bed linens. 14x + 60y ≤ 300
b. To graph the inequality, solve for x first. Graph. Then use the test point (0, 0) to determine the shading. The point (0, 0) results in 0≤ 5 which is true. Shade
the side that includes the origin. One possible solution would be to buy 5 gallons of paint Calculate how many bed linen sets must be purchased when
5 gallons of paint are bought. Thus when 5 gallons of paint are purchased, 3 bed linen sets can be purchased. Use a graph to solve each inequality.
33. −6x − 8 ≥ −4
SOLUTION: Graph
the boundary,
is the related function. Replace the inequality sign with an equal sign and solve for x. Page 6
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5-6 Graphing
Inequalities in Two Variables
Thus when 5 gallons of paint are purchased, 3 bed linen sets can be purchased. Use a graph to solve each inequality.
33. −6x − 8 ≥ −4
SOLUTION: 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 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
.
36. −4x − 4 ≤ −6
SOLUTION: Graph the boundary, which is the related function. Replace the inequality sign with an equal sign and solve for x.
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solution is
.
5-6 Graphing
Inequalities in Two Variables
36. −4x − 4 ≤ −6
SOLUTION: 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 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
.
Graph each inequality. Determine which of the ordered pairs are part of the solution set for each
inequality.
39. x < −4; {(2, 1), (−3, 0), (0, −3), (−5, −5), (−4, 2)}
SOLUTION: y < –4
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Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
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solution is
.
5-6 Graphing
Inequalities in Two Variables
Graph each inequality. Determine which of the ordered pairs are part of the solution set for each
inequality.
39. x < −4; {(2, 1), (−3, 0), (0, −3), (−5, −5), (−4, 2)}
SOLUTION: 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).
To determine which of the ordered pairs are part of the solution set, plot them on the graph. Because (–5, –5) is in the shaded half–plane, it is part of the solution set.
42. −3x + 5y < 10; {(3, −1), (1, 1), (0, 8), (−2, 0), (0, 2)}
SOLUTION: Solve for y in terms of x.
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Because the inequality involves <, graph the boundary using a dashed line. Choose (0, 0) as a test point.
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5-6 Graphing
Inequalities
in shaded
Two Variables
Because (–5,
half–plane, it is part of the solution set.
–5) is in the
42. −3x + 5y < 10; {(3, −1), (1, 1), (0, 8), (−2, 0), (0, 2)}
SOLUTION: 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).
To determine which of the ordered pairs are part of the solution set, plot them on the graph. Because (3, –1), (1, 1), and (2, 0) are in the shaded half–plane, they are part of the solution set.
45. MULTIPLE REPRESENTATIONS Use inequalities A and B to investigate graphing compound inequalities on a
coordinate plane.
A. 7(y + 6) ≤ 21x + 14
B. −3y
≤ 3x
−12 by Cognero
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a. NUMERICAL Solve each inequality for y.
b. GRAPHICAL Graph both inequalities on one graph. Shade the half–plane that makes A true in red. Shade the
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.
45. MULTIPLE REPRESENTATIONS Use inequalities A and B to investigate graphing compound inequalities on a
coordinate plane.
A. 7(y + 6) ≤ 21x + 14
B. −3y ≤ 3x −12
a. NUMERICAL Solve each inequality for y.
b. GRAPHICAL 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.
c. VERBAL What does the overlapping region represent?
SOLUTION: a. For equation A:
For equation B:
b.
c. The overlapping region represents the solutions that make both A and B true.
46. ERROR ANALYSIS Reiko and Kristin are solving
by graphing. Is either of them correct? Explain your
reasoning.
SOLUTION: Kristin used a test point located on the line and shaded the incorrect half-plane. Reiko is correct.
48. REASONING Explain why a point on the boundary should not be used as a test point.
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SOLUTION: The test point is used to determine which half–plane is the solution set for the inequality. A test point on the boundary
SOLUTION: 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.
48. REASONING Explain why a point on the boundary should not be used as a test point.
SOLUTION: 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.
50. WRITING IN MATH Summarize the steps to graph an inequality in two variables.
SOLUTION: 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.
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