6–7
Graphing Inequalities in Two Variables
BUILD YOUR VOCABULARY (pages 125–126)
MAIN IDEAS
The region of the graph of an inequality on one side of the
• Graph inequalities on
is called a half-plane.
the coordinate plane.
• Solve real-world
problems involving
linear inequalities.
An
defines the boundary or edge for each
half-plane.
KEY CONCEPT
Half-Planes and
Boundaries Any line in
the plane divides the
plane into two regions
called half-planes.
The line is called the
boundary of each of the
two half-planes.
EXAMPLE
Graph an Inequality
Graph 2y - 4x > 6.
Step 1 Solve for y in terms of x.
2y - 4x > 6
2y - 4x +
Original Inequality
>
+6
Add
to each side.
2y > 4x + 6
Simplify.
2y
4x + 6
_
> __
Divide each side by 2.
2
y>
Simplify.
Step 2 Graph y = 2x + 3.
Since y > 2x + 3 does not include values when
y = 2x + 3, the boundary is
in
the solution set. The boundary should be drawn as a
.
y
Step 3 Select a point in one of the
half-planes and test it.
Let’s use (0, 0).
y > 2x + 3
Original inequality
0 > 2(0) + 3
x = 0, y = 0
0>3
False
Since the statement is false, the
the origin is
half-plane.
140
Glencoe Algebra 1
y 2x 3
O
containing
part of the solution. Shade the other
x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
6–7
Check Test a point in the other half-plane, for example, (-3, 1).
y > 2x + 3
Original inequality
1 > 2(-3) + 3
x = -3, y = 1
1 > -3 ✓
Since the statement is true, the half-plane containing (-3, 1)
should be
REMEMBER IT
.
Check Your Progress Graph y - 3x < 2.
A dashed line
indicates that the
boundary is not part of
the solution set. A solid
line indicates that the
boundary line is part of
the solution set.
y
O
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
EXAMPLE
ORGANIZE IT
In Lesson 6-7 of your
Foldable, explain how
to check the solution
to an inequality in two
variables.
x
Write and Solve an Inequality
JOURNALISM Lee Cooper writes and edits short articles
for a local newspaper. It generally takes her an hour to
write an article and about a half-hour to edit an article.
If Lee works up to 8 hours a day, how many articles can
she write and edit in one day?
Step 1 Let x equal the number of articles Lee can write. Let y
equal the number of articles that Lee can edit. Write an
open sentence representing the situation.
Number of
articles
she can write
plus
_1 hour
2
times
number of
articles
she can edit
is up to
8 hours.
Solving Linear
Inequalities
+
×
8
Step 2 Solve for y in terms of x.
1
x+_
y≤8
Original inequality
2
1
x+_
y-
≤
2
+8
≤ -x + 8
1
(2) _
y ≤ 2(-x + 8)
2
y≤
Subtract
from each side.
Simplify.
Multiply each side by 2.
Simplify.
Glencoe Algebra 1
141
6–7
Step 3 Since the open sentence includes the equation, graph
y = -2x + 16 as a
line. Test a
in one
of the half-planes, for example, (0, 0). Shade the halfplane containing (0, 0) since 0 ≤ -2(0) + 16 is true.
18
16
14
12
10
8
6
4
2
O
y
2 4 6 8 10 12 14 16 18 x
Step 4 Examine the situation
• Lee cannot work a negative number of hours. Therefore, the
domain and range contain only
numbers.
• Lee only wants to count articles that are completely written
or completely edited. Thus, only points in the half-plane
whose x- and y-coordinates are
possible solutions.
• One solution is (2, 3). This represents
written articles
edited articles.
Check Your Progress You offer to go to the local deli and
pick up sandwiches for lunch. You have $30 to spend. Chicken
sandwiches cost $3.00 and tuna sandwiches are $1.50 each.
How many sandwiches can you purchase for $30?
HOMEWORK
ASSIGNMENT
Page(s):
Exercises:
142
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
and
numbers are