128 CHAPTER 1
Linear Equations and Inequalities
1.8 Linear Inequalities in Two Variables
OBJECTIVES
1
Determine Whether an Ordered
Pair Is a Solution to a Linear
Inequality
2
Graph Linear Inequalities
3
Solve Problems Involving Linear
Inequalities
Preparing for Linear Inequalities in Two Variables
Before getting started, take this readiness quiz. If you get a problem wrong, go back to the section
cited and review the material.
P1. Determine whether x = 4 satisfies the
inequality 3x + 1 Ú 7.
P2. Solve the inequality: -4x - 3 7 9
[Section 1.4, pp. 85–88]
[Section 1.4, pp. 85–88]
In Section 1.4, we solved inequalities in one variable. In this section, we discuss linear
inequalities in two variables.
1
Determine Whether an Ordered Pair Is a Solution
to a Linear Inequality
Linear inequalities in two variables are inequalities in one of the forms
Ax + By 6 C
Ax + By 7 C
Ax + By … C
Ax + By Ú C
where A, B, and C are real numbers and A and B are not both zero.
If we replace the inequality symbol with an equal sign, we obtain the equation of a
line, Ax + By = C. The line separates the xy-plane into two regions, called halfplanes. See Figure 56.
Figure 56
y
Ax ⫹ By ⫽ C
x
A linear inequality in two variables x and y is satisfied by an ordered pair (a, b) if,
when x is replaced by a and y is replaced by b, a true statement results.
EXAMPLE 1 Determining Whether an Ordered Pair Is a Solution
to a Linear Inequality In Two Variables
Determine which of the following ordered pairs are solutions to the linear inequality
3x + y 6 7.
(a) (2, 4)
(b) 1-3, 12
(c) (1, 3)
Solution
(a) Let x = 2 and y = 4 in the inequality. If a true statement results, then (2, 4)
is a solution to the inequality.
3x + y 6 7
x = 2, y = 4:
?
3122 + 4 6 7
?
6 + 4 6 7
?
10 6 7 False
Preparing for...Answers P1. Satisfies
P2. {x | x 6 -3} or ( - q , -3)
Because 10 is not less than 7, the statement is false, so (2, 4) is not a solution to
the inequality.
Section 1.8
Linear Inequalities in Two Variables 129
(b) Let x = -3 and y = 1 in the inequality. If a true statement results, then
1-3, 12 is a solution to the inequality.
3x + y 6 7
x = -3, y = 1:
?
31-32 + 1 6 7
?
-9 + 1 6 7
?
-8 6 7
True
Because -8 is less than 7, the statement is true, so 1-3, 12 is a solution to
the inequality.
(c) Let x = 1 and y = 3 in the inequality.
3x + y 6 7
x = 1, y = 3:
?
3112 + 3 6 7
?
3 + 3 6 7
?
6 6 7 True
Because 6 is less than 7, the statement is true, so (1, 3) is a solution to the
inequality.
Quick
1. If we replace the inequality symbol in Ax + By 7 C with an equal sign, we
obtain the equation of a line, Ax + By = C. The line separates the xy-plane
into two regions, called
.
2. Determine which of the following ordered pairs are solutions to the linear
inequality -2x + 3y Ú 3.
(a) (4, 1)
Work Smart
To review the distinction between
strict and nonstrict inequalities, turn
back to page 17.
2
(b) 1-1, 22
(c) (2, 3)
(d) (0, 1)
Graph Linear Inequalities
Now that we know how to determine whether an ordered pair is a solution to a linear
inequality in two variables, we are prepared to graph linear inequalities in two variables.
A graph of a linear inequality in two variables x and y consists of all points (x, y) whose
coordinates satisfy the inequality.
The graph of any linear inequality in two variables may be obtained by graphing the
equation corresponding to the inequality, using dashes if the inequality is strict (⬍ or
⬎) and a solid line if the inequality is nonstrict ( … or Ú ). This graph will separate the
xy-plane into two half-planes. In each half-plane either all points satisfy the inequality
or no points satisfy the inequality. So the use of a single test point is all that is required
to obtain the graph of a linear inequality in two variables.
EXAMPLE 2 How to Graph a Linear Inequality in Two Variables
Graph the linear inequality: 3x + y 6 7
Step-by-Step Solution
Step 1: We replace the inequality symbol with an equal sign
and graph the corresponding line. If the inequality is strict
( 6 or 7 ), graph the line as a dashed line. If the inequality is
nonstrict ( … or Ú ), graph the line as a solid line.
We replace 6 with = to obtain 3x + y = 7. We graph
the line 3x + y = 7 1y = -3x + 72 using a dashed line
because the inequality is strict. See Figure 57(a) on the
next page.
130 CHAPTER 1
Linear Equations and Inequalities
3x + y 6 7
Step 2: We select any test point that is not on the line
and determine whether the test point satisfies the
inequality. When the line does not contain the origin, it
is usually easiest to choose the origin, (0, 0), as the
test point.
Figure 57
?
3102 + 0 6 7
Test Point: (0, 0):
?
0 6 7 True
Because 0 is less than 7, the point (0, 0) satisfies the
inequality. Therefore, we shade the half-plane containing
the point (0, 0). See Figure 57(b). The shaded region
represents the solution to the linear inequality.
y
y
9
9
(0, 7)
6
(0, 7)
6
3x ⫹ y ⫽ 7
3x ⫹ y ⫽ 7
3
3
⫺3
3
⫺3 (0, 0)
x
3
x
⫺3
⫺3
(a)
(b)
Only one test point is needed to obtain the graph of the inequality. Why? Consider
the inequality 3x + y 6 7 presented in Examples 1 and 2. We notice that A12, 42 does
not satisfy the inequality, while B1-3, 12 and C11, 32 do satisfy the inequality. Notice
that point A is not in the shaded region of Figure 57(b), while points B and C are in the
shaded region. So, if a point does not satisfy the inequality, then none of the points in the
half-plane containing that point satisfy the inequality. If a point does satisfy the inequality, then all the points in the half-plane containing the point satisfy the inequality.
Below we summarize the steps for graphing a linear inequality in two variables.
STEPS FOR GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Work Smart
An alternative to using test points is
to solve the inequality for y. If the
inequality is of the form y 7 or y Ú,
shade above the line. If the
inequality is of the form y 6 or y … ,
shade below the line.
Step 1: Replace the inequality symbol with an equal sign and graph the resulting
equation. If the inequality is strict (⬍ or ⬎), use dashes to graph the line; if
the inequality is nonstrict ( … or Ú ) use a solid line. The graph separates
the xy-plane into two half-planes.
Step 2: Select a test point P that is not on the line (that is, select a test point in one
of the half-planes).
(a) If the coordinates of P satisfy the inequality, then shade the half-plane
containing P.
(b) If the coordinates of P do not satisfy the inequality, then shade the
half-plane that does not contain P.
Quick
3. True or False: The graph of a linear inequality is a line.
4. True or False: In a graph of a linear inequality in two variables with a strict
inequality, the line separating the two half-planes should be dashed.
In Problems 5 and 6, graph each linear inequality.
5. y 6 -2x + 3
6. 6x - 3y … 15
Section 1.8 Linear Inequalities in Two Variables 131
EXAMPLE 3 Graphing a Linear Inequality in Two Variables
Graph the linear inequality: y Ú
1
x
2
Solution
Work Smart
Do not use (0, 0) as a test point for
equations of the form Ax + By = 0
because the graph of this equation
contains the origin.
1
We replace the inequality symbol with an equal sign to obtain y = x. We graph the
2
1
line y = x using a solid line because the inequality is nonstrict. See Figure 58(a).
2
Now, we select any test point that is not on the line and determine whether the
test point satisfies the inequality. Because the line contains the origin, we decide to
use (2, 0) as the test point.
1
y Ú x
2
? 1
Test Point: (2, 0): 0 Ú # 2
2
?
0 Ú 1
False
Because 0 is not greater than 1, the point (2, 0) does not satisfy the inequality. Therefore,
we shade the half-plane that does not contain (2, 0). See Figure 58(b). The shaded region
represents the solution to the linear inequality.
Notice the inequality is of the form y Ú , so we shade above the line.
Figure 58
y
y
3
3
⫺3
3
x
⫺3
(a)
⫺3
(2, 0) 3
x
⫺3
(b)
Quick
7. Graph the linear inequality: 2x + y 6 0
3 Solve Problems Involving Linear Inequalities
Linear inequalities involving two variables can be used to solve problems in areas such
as nutrition, manufacturing, or sales. Let’s look at an application of linear inequalities
from nutrition.
EXAMPLE 4 Saturated Fat Intake
Randy really enjoys Wendy’s Junior Cheeseburgers and Biggie French Fries.
However, he knows that his intake of saturated fat during lunch should not exceed
16 grams. Each Junior Cheeseburger contains 6 grams of saturated fat and each
Biggie Fries contains 3 grams of saturated fat. (SOURCE: wendys.com)
(a) Write a linear inequality that describes Randy’s options for eating at
Wendy’s. That is, write an inequality that represents all the combinations
of Junior Cheeseburgers and Biggie Fries that Randy can order.
(b) Can Randy eat 2 Junior Cheeseburgers and 1 Biggie Fry during lunch
and stay within his allotment of saturated fat?
(c) Can Randy eat 3 Junior Cheeseburgers and 1 Biggie Fry during lunch
and stay within his allotment of saturated fat?
Solution
(a) We are going to use the first three steps in the problem-solving strategy
given in Section 1.2 on page 61 to help us develop the linear inequality.
132 CHAPTER 1
Linear Equations and Inequalities
Step 1: Identify We want to determine the number of Junior Cheeseburgers
and Biggie Fries Randy can eat while not exceeding 16 grams of saturated fat.
Step 2: Name the Unknowns Let x represent the number of Junior Cheeseburgers that Randy eats and let y represent the number of Biggie Fries Randy eats.
Step 3: Translate If Randy eats one Junior Cheeseburger, then he will consume 6 grams of saturated fat. If he eats two, then he will consume 12 grams
of saturated fat. In general, if he eats x Junior Cheeseburgers, he will consume 6x grams of saturated fat. Similar logic for the Biggie Fries tells us that
if Randy eats y Biggie Fries, he will consume 3y grams of saturated fat. The
words “cannot exceed” imply a … inequality. Therefore, a linear inequality
that describes Randy’s options for eating at Wendy’s is
6x + 3y … 16
The Model
(b) Letting x = 2 and y = 1, we obtain
?
6122 + 3112 … 16
?
15 … 16 True
Because the inequality is true, Randy can eat 2 Junior Cheeseburgers and
1 Biggie Fry and remain within the allotment of 16 grams of saturated fat.
(c) Letting x = 3 and y = 1, we obtain
?
6132 + 3112 … 16
?
21 … 16 False
Because the inequality is false, Randy cannot eat 3 Junior Cheeseburgers and
1 Biggie Fry and remain within the allotment of 16 grams of saturated fat.
Quick
8. Avery is on a diet that requires that he consume no more than 800 calories for
lunch. He really enjoys Wendy’s Chicken Breast filet and Frosties. Each Chicken
Breast filet contains 430 calories and each Frosty contains 330 calories.
(a) Write a linear inequality that describes Avery’s options for eating at Wendy’s.
(b) Can Avery eat 1 Chicken Breast filet and 1 Frosty and stay within his
allotment of calories?
(c) Can Avery eat 2 Chicken Breast filets and 1 Frosty and stay within his
allotment of calories?
1.8 EXERCISES
1–8. are the Quick
s that follow each EXAMPLE
Building Skills
In Problems 9–12, determine whether the given points are solutions
to the linear inequality. See Objective 1.
9. x + 3y 6 6
(a) (0, 1)
(b) 1-2, 42
(c) 18, -12
11. -3x + 4y Ú 12
(a) 1-4, 22
(b) (0, 2)
(c) (0, 3)
10. 2x + y 7 -3
(a) 12, -12
(b) 11, -32
(c) 1-5, 42
12. 2x - 5y … 2
(a) 11, 02
(b) (3, 0)
(c) 11, 22
In Problems 13–32, graph each inequality. See Objective 2.
13. y 7 3
14. y 6 -2
15. x Ú -2
16. x 6 7
17. y 6 5x
18. y Ú
19. y 7 2x + 3
20. y 6 -3x + 1
21. y …
1
x - 5
2
2
x
3
4
22. y Ú - x + 5
3
23. 3x + y … 4
24. -4x + y Ú -5
25. 2x + 5y … -10
26. 3x + 4y Ú 12
27. -4x + 6y 7 24
28. -5x + 3y 6 30