Transcript (Graphing Linear Inequalities in Two Variables).

Graphing Linear Inequalities in Two
Variables
An example of a linear inequality in two
variables is x - 3y ≤ 6. The solution of an
inequality in two variables, x and y, is an
ordered pair (x, y) that produces a true
statement when substituted into the inequality.
Which ordered pair is NOT a solution of x - 3y ≤ 6?
A. (0,0)
B. (6,-1)
C. (10, 3)
D. (-1,2)
Substitute each point into the inequality.
If the statement is true then it is a solution.
x - 3y ≤ 6
(0) – 3(0) ≤ 6
0≤6
True, therefore
(0,0) is a solution.
Expressions of the type x + 2y ≤ 8 and 3x – y > 6
are called linear inequalities in two variables.
A solution of a linear inequality in two variables is an
ordered pair (x, y) which makes the inequality true.
Example: (1, 3) is a solution to x + 2y ≤ 8
since (1) + 2(3) ≤ 8
7 ≤ 8. (Yes, this is true.)
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The graph of an inequality in two variables is
the set of points that represent all solutions
of the inequality.
The BOUNDARY LINE of a linear inequality
divides the coordinate plane into two
HALF-PLANES. Only one half-plane contains
the points that represent the solutions to the
inequality.
The solution set, or feasible set, of a linear inequality
in two variables is the set of all solutions.
y
Example:
The solution set for x + 2y ≤ 8
is the shaded region.
2
2
x
The solution set is a half-plane. It consists of the line
x + 2y ≤ 8 and all the points below and to its left.
The line is called the boundary line of the half-plane.
Some solutions in the solution set are (0,0), (2, -2), and (-4, 2).
(6, 4) is not in the solution set!
If the inequality is ≤ or ≥ ,
the boundary line is solid;
its points are solutions.
y
3x – y = 2
x
3x – y < 2
3x – y > 2
Example: The boundary line of the
solution set of 3x – y ≥ 2 is solid.
If the inequality is < or >,
the boundary line is dotted;
its points are not solutions.
y
Example: The boundary line of the
solution set of x + y < 2 is dotted.
x
A test point can be selected to determine which side of the
half-plane to shade. Pick any point that is not on your line.
Example: For 2x – 3y ≤ 18
graph the boundary line.
y
(0, 0)
Use (0, 0) as a test point.
-2
x
2
2(0) – 3(0) ≤ 18
0 ≤ 18 Yes/True
Shade towards your test point/Include your test point in
the shading!
The solution set/feasible set, is the set of all solutions in
the shaded region.
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To graph the solution set/feasible set for a linear inequality:
Step 1. Graph the boundary line. (Remember to
check if your line is a dotted line or a solid line.)
Step 2. Select a test point, not on the boundary line,
and determine if it is a solution.
Step 3. Shade a half-plane.
Example: Graph the solution set for x – y > 2.
Step 1.) Graph the boundary line x – y = 2 as a dotted line.
Step 2.) Select a test point
not on the line, say (0, 0).
x–y>2
y
(0, 0)
x
(2, 0)
(0) – (0) > 2
(0, -2)
0 > 2 No/False!
Step 3.) Since this is a not a solution, shade in the
half-plane not containing (0, 0). Or, shade away from
your test point.
The solution set/feasible set, is the set of all solutions
in the shaded region.
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Solution sets for inequalities with only one variable can be graphed in
the same way. Example: Graph the solution set for x < - 2.
Step 1.) Graph the solution set for x < -2. y
Step 2.) Select a test point not
on the line, say (0, 0).
x<-2
0 < - 2 No/False!
4
x
-4
4
-4
Step 3.) Since this is a not a solution, shade in the
half-plane not containing (0, 0). Or, shade away
from your test point.
The solution set/feasible set, is the set of all
solutions in the shaded region.
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Example: Graph the solution set for x ≥ 4.
Step 1.) Graph the solution set for x ≥ 4. y
Step 2.) Select a test point not on the line, 4
say (0, 0).
x≥4
-4
(0) ≥ 4 No/False!
x
4
-4
Since this is a not a solution, shade in the halfplane not containing (0, 0). Or, shade away from
your test point.
The solution set/feasible set, is the set of all solutions in
the shaded region.
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Graph the inequality
STEP 1
Graph the equation
y  4x  3
STEP 2
Test (0,0) in the
original inequality.
y  4x  3
0  4(0)  3
True
y > 4x - 3.
STEP 3
Shade the half-plane
that contains the point
(0,0), because (0,0) is a
solution to the
inequality.
Graph the inequality
STEP 1
Graph the equation
x  2y  0
x + 2y ≤ 0.
STEP 2
Test (1,0) in the
original inequality.
x  2y  0
1  2( 0)  0
1 0
False
STEP 3
Shade the half-plane
that does not contain
the point (1,0),
because (1,0) is not a
solution to the
inequality.
Graph the inequality
STEP 1
Graph the equation
1  x  y
STEP 2
Test (0,0) in the
original inequality.
1  x  y
 1  (0)  (0)
1 0
True
-1 ≤ x + y.
STEP 3
Shade the half-plane
that contains the point
(0,0), because (0,0) is a
solution to the
inequality.
Graph an Inequality
Graph the inequality
STEP 1
Graph the equation
2 x  3 y  12
x – 3y ≤ 12.
STEP 2
Test (0,0) in the
original inequality.
2 x  3 y  12
2(0)  3(0)  12
0  12
True
STEP 3
Shade the half-plane
that contains the point
(0,0), because (0,0) is a
solution to the
inequality.
Graph the inequality
STEP 1
Graph the equation
y  3
STEP 2
Test (0,0) in the
original inequality.
Use only the ycoordinate, because
the inequality does
not have a x-variable.
y  3
( 0 )  3
True
y ≥ -3.
STEP 3
Shade the half-plane that
contains the point (0,0),
because (0,0) is a solution
to the inequality.
Graph the inequality
STEP 1
Graph the equation
x  1
STEP 2
Test (0,0) in the
original inequality.
Use only the ycoordinate, because
the inequality does
not have a x-variable.
x  1
( 0 )  1
False
x ≤ -1.
STEP 3
Shade the half-plane that
does not contain the
point (0,0), because (0,0)
is not a solution to the
inequality.
Write an inequality for each graph.
a.
b.
y  x2
1
y  x3
3
Write an inequality for each graph.
c.
d.
x3
y  2

Study Guide 6-6 (In-Class)
Skills Practice/Practice Worksheet 6-6

Chapter 6 Test on

d. Suppose your budget for a party allows you to
spend no more than $12 on peanuts and cashews.
Peanuts cost $2/lb and cashews cost $4/lb. Find
three possible combinations of peanuts and
cashews you can buy.
x = number of pounds of peanuts
y = number of pounds of cashews
2x + 4y ≤ 12
A solution of a system of linear inequalities is an
ordered pair that satisfies all the inequalities.
x  y  8
Example: Find a solution for the system 
.
2 x  y  7
(5, 4) is a solution of x + y > 8.
(5, 4) is also a solution of 2x – y ≤ 7.
Since (5, 4) is a solution of both inequalities in the
system, it is a solution of the system.
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The set of all solutions of a system of linear inequalities
is called its solution set.
To graph the solution set for a system of linear
inequalities in two variables:
1. Shade the half-plane of solutions for each
inequality in the system.
2. Shade in the intersection of the half-planes.
Example:
x  y  8
Graph the solution set for the system 
2 x  y  7
y
Graph the solution set for
x + y > 8.
Graph the solution set for
2x – y ≤ 7.
2
2
The intersection of these two
half-planes is the wedge-shaped
region at the top of the
diagram.
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rights reserved.
x
Example: Graph the solution set for the system of
linear inequalities: 2 x  3 y  12

 2 x  3 y  6
y
-2x + 3y ≥ 6
Graph the two half-planes.
2
The two half-planes do not
intersect;
therefore, the solution set is
the empty set.
x
2
2x – 3y ≥ 12
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rights reserved.
Example: Graph the solution set for the linear system.
(1) 2 x  3 y  3
(2) 
6 x  y  1

(3)  x  2
(4) 
 y  1
(2) y
(1)
4
x
-4
4
Graph each linear
inequality.
(4)
-4
(3)
The solution set is the intersection of all the half-planes.
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Mifflin Company, Inc. All
rights reserved.