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Graphing
Inequalities
inin
Graphing
Inequalities
11-6
11-6Two
Variables
Two
Variables
Warm Up
Problem of the Day
Lesson Presentation
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Warm Up
Find each equation of direct variation, given
that y varies directly with x.
1. y is 18 when x is 3.
y = 6x
1x
y
=
2. x is 60 when y is 12.
5
3. y is 126 when x is 18. y = 7x
4. x is 4 when y is 20.
Pre-Algebra
y = 5x
Graphing Inequalities in
11-6 Two Variables
Problem of the Day
The circumference of a pizza varies
directly with its diameter. If you graph
that direct variation, what will the slope
be? 
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Today’s Learning Goal Assignment
Learn to graph
inequalities on the
coordinate plane.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Vocabulary
boundary line
linear inequality
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
A graph of a linear equation separates
the coordinate plane into three parts:
the points on one side of the line, the
points on the boundary line, and the
points on the other side of the line.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
When the equality symbol is replaced
in a linear equation by an inequality
symbol, the statement is a linear
inequality. Any ordered pair that
makes the linear inequality true is a
solution.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Additional Example 1A: Graphing Inequalities
Graph each inequality.
A. y < x – 1
First graph the boundary line y = x – 1. Since
no points that are on the line are solutions of
y < x – 1, make the line dashed. Then
determine on which side of the line the
solutions lie.
(0, 0)
Test a point not on the line.
y<x–1
?
0<0–1
?
0 < –1
Substitute 0 for x and 0 for y.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Additional Example 1A Continued
Since 0 < –1 is not true, (0, 0) is not a solution of
y < x – 1. Shade the side of the line that does not
include (0, 0).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1A
Graph each inequality.
A. y < x – 4
First graph the boundary line y = x – 4. Since
no points that are on the line are solutions of
y < x – 4, make the line dashed. Then
determine on which side of the line the
solutions lie.
(0, 0)
Test a point not on the line.
y<x–4
?
0<0–4
?
0 < –4
Substitute 0 for x and 0 for y.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1A Continued
Since 0 < –4 is not true, (0, 0) is not a solution of
y < x – 4. Shade the side of the line that does not
include (0, 0).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Additional Example 1B: Graphing Inequalities
B. y  2x + 1
First graph the boundary line y = 2x + 1. Since
points that are on the line are solutions of
y  2x + 1, make the line solid. Then shade the
part of the coordinate plane in which the rest of
the solutions of y  2x + 1 lie.
(0, 4)
Choose any point not on the line.
y ≥ 2x + 1
?
4≥0+1
Pre-Algebra
Substitute 0 for x and 4 for y.
Graphing Inequalities in
11-6 Two Variables
Additional Example 1B Continued
Since 4  1 is true, (0, 4) is a solution of y  2x + 1.
Shade the side of the line that includes (0, 4).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1B
B. y > 4x + 4
First graph the boundary line y = 4x + 4. Since
points that are on the line are solutions of
y  4x + 4, make the line solid. Then shade the
part of the coordinate plane in which the rest of
the solutions of y  4x + 4 lie.
(2, 3)
Choose any point not on the line.
y ≥ 4x + 4
?
3≥8+4
Pre-Algebra
Substitute 2 for x and 3 for y.
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1B Continued
Since 3  12 is not true, (2, 3) is not a solution of
y  4x + 4. Shade the side of the line that does
not include (2, 3).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Additional Example 1C: Graphing Inequalities
C. 2y + 5x < 6
First write the equation in slope-intercept form.
2y + 5x < 6
2y < –5x + 6
y < –5 x + 3
2
Subtract 5x from both sides.
Divide both sides by 2.
Then graph the line y = – 5x + 3. Since points that
2
are on the line are not solutions of y < – 5 x + 3,
2
make the line dashed. Then determine on which
side of the line the solutions lie.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Additional Example 1C Continued
(0, 0)
y < –5
2x + 3
?
0<0+3
Choose any point not on the line.
?
0<3
Since 0 < 3 is true, (0, 0) is a
solution of y < – 5 x + 3.
2
Shade the side of the line
that includes (0, 0).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1C
C. 3y + 4x  9
First write the equation in slope-intercept form.
3y + 4x  9
3y  –4x + 9
y  –4x + 3
3
Subtract 4x from both sides.
Divide both sides by 3.
Then graph the line y = – 4x + 3. Since points that
3
are on the line are solutions of y  – 4 x + 3,
3
make the line solid. Then determine on which side
of the line the solutions lie.
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Try This: Example 1C Continued
(0, 0)
y  –4 x + 3
3
?
00+3
Choose any point not on the line.
?
03
Since 0  3 is not true, (0, 0) is
not a solution of y  – 4 x + 3.
3
Shade the side of the line that
does not include (0, 0).
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Lesson Quiz
Graph each inequality.
1. y < –1 x + 4
3
2. 4y + 2x > 12
Tell whether the given ordered pair is a
solution of each inequality.
3. y < x + 15 (–2, 8)
4. y  3x – 1 (7, –1)
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
1. y < –1 x + 4
3
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
2. 4y + 2x > 12
Pre-Algebra
Graphing Inequalities in
11-6 Two Variables
Tell whether the given ordered pair is a
solution of each inequality.
3. y < x + 15 (–2, 8) yes
4. y  3x – 1 (7, –1)
Pre-Algebra
no