Download Topic 7.6

Topic 7.6
Systems of
Inequalities
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Topic 7.6,
1.1, Slide 1
OBJECTIVES
1. Determine if an Ordered Pair Is a Solution to an
Inequality in Two Variables
2. Graphing a Liner Inequality in Two Variables
3. Graphing a Nonlinear Inequality in Two Variables
4. Determining If an Ordered Pair Is a Solution to a
System of Inequalities in Two Variables
5. Graphing a System of Linear Inequalities in Two
Variables
6. Graphing a System of Nonlinear Inequalities in Two
Variables
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Topic 7.6,
1.1, Slide 2
Determine if an Ordered Pair Is a
Solution to an Inequality in Two
Variables
EXAMPLE
Determine if the given ordered pair is a solution to the inequality −2 x + y 2 ≤ 4.
a. (2, −1)
b. (−2, 0)
−2 ( 2 ) + ( −1) ≤ 4
2
−4 + 1 ≤ 4
−3 ≤ 4
true
−2 ( −2 ) + ( 0 ) ≤ 4
 1 7
c.  − , 
 2 2
2
4+0 ≤ 4
4≤4
true
2
 1 7
−2  −  +   ≤ 4
 2 2
1+
49
≤4
4
53
≤4
4
false
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Topic 7.6,
1.1, Slide 3
Graphing a Linear Inequality in Two
Variables
Definition: Linear Inequality in Two Variables
A linear inequality in two variables is an inequality that can be written
In the form Ax + By = C, where A, B, and C are real numbers, and A
and B are not both equal to zero.
Note: The inequality symbol "<" can be replaced with >, ≤ , or ≥ .
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Topic 7.6,
1.1, Slide 4
Graphing a Linear Inequality in Two
Variables
The line x + y = 3 divides the coordinate plane into two half-planes, an
upper half-plane and a lower half-plane. The upper half-plane is shaded
blue and the lower half-plane is shaded red.
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Topic 7.6,
1.1, Slide 5
Graphing a Linear Inequality in Two
Variables
Let’s now examine some points that lie in each half-plane to see which
ones satisfy the inequalities
x + y > 3 or x + y < 3.
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Topic 7.6,
1.1, Slide 6
Graphing a Linear Inequality in Two
Variables
The equation x + y = 3 acts as a boundary line that separates the
solutions of the two inequalities x + y > 3 and x + y < 3.
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Topic 7.6,
1.1, Slide 7
Graphing a Linear Inequality in Two
Variables
Steps for Graphing Linear Inequalities in Two Variables
Step 1. Find the boundary line for the inequality by replacing the inequality
symbol with an equal sign and graphing the resulting equation. If the
inequality is strict, graph the boundary line using a dashed line. If the
inequality is non-strict, graph the boundary line using a solid line.
Step 2. Choose a test point that does not belong to the boundary line and
determine if it is a solution to the inequality.
Step 3. If the test point is a solution to the inequality, then shade the half-plane
that contains the test point. If the test point is not a solution to the
inequality, then shade the half-plane that does not contain the test point.
The shaded area represents the set of all ordered-pair solutions to the
inequality.
When graphing a linear inequality in two variables, using a dashed line is similar to
using an open circle when graphing a linear inequality in one variable on a number
line. Likewise, using a solid line is similar to using a solid circle
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Topic 7.6,
1.1, Slide 8
Graphing a Linear Inequality in Two
Variables
EXAMPLE
Graph the inequality
x − 2 y ≥ 4.
Step 1. The boundary line is x - 2y = 4. We graph the line as a solid line because
the inequality is non-strict.
Step 2. We choose the test point (0, 0) and check to see if it satisfies the
inequality.
x − 2y ≥ 4
0 − 2(0) ≥ 4
0≥4
false
The point (0, 0) is not a solution to the
inequality.
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2012,
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Topic 7.6,
1.1, Slide 9
Graphing a Linear Inequality in Two
Variables
EXAMPLE
Graph the inequality
x − 2 y ≥ 4.
Step 3. Because the test point is not a solution to the inequality, we shade the
half-plane that does not contain (0, 0).
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Topic 7.6,
1.1, Slide 10
Graphing a Nonlinear Inequality in
Two Variables
Steps for Graphing Nonlinear Inequalities in Two Variables
Step 1. Replace the inequality symbol with an equal sign and graph the resulting
equation. If the inequality is strict, sketch the graph using dashes. If the
inequality is non-strict, sketch the graph using a solid curve. This graph
divides the coordinate plane into two or more regions.
Step 2. Choose one test point that does not belong to the graph of the equation
from step 1 and determine if it is a solution to the inequality.
Step 3. If the test point is a solution to the inequality, shade the region that
contains the test point. If the test point is not a solution to the inequality,
shade the region that does not include the test point.
In order to be certain that all of the solution set has been identified, it may be
necessary to choose more than one test point when the graph of a nonlinear
inequality in two variables divides the coordinate plane into multiple regions.
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Topic 7.6,
1.1, Slide 11
Graphing a Nonlinear Inequality in
Two Variables
EXAMPLE
Graph the inequality x 2 + y 2 ≥ 9.
Step 1. The inequality is non-strict, so we graph the equation x2 + y2 = 9 (which
is a circle centered at the origin having a radius of 3 units) using a continuous solid
curve.
Step 2. We choose the test point (0, 0), which lies inside the circle.
x2 + y 2 ≥ 9
(0)2 + (0)2 ≥ 9
0≥9
false
The point (0, 0) that lies inside the circle is
not a solution to the inequality.
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Topic 7.6,
1.1, Slide 12
Graphing a Nonlinear Inequality in
Two Variables
EXAMPLE
Graph the inequality x 2 + y 2 ≥ 9.
Step 3. Because the test point (0, 0) is not a solution to the inequality, we shade
the region that lies outside of the circle.
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Topic 7.6,
1.1, Slide 13
Determining If an Order Pair Is a
Solution to a System of Inequalities
in Two Variables
Definition: Systems of Linear Inequalities in
Two Variables
A system of inequalities in two variables is the collection of two
or more inequalities in two variables considered together. If all
inequalities are linear, then the system is a system of linear
inequalities in two variables. If at least one inequality is nonlinear,
then the system is a system of nonlinear inequalities in two
variables.
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Topic 7.6,
1.1, Slide 14
Determining If an Order Pair Is a
Solution to a System of Inequalities
in Two Variables
EXAMPLE
Determine which ordered pairs are solutions to the given system.
2x − 3 y ≤ 9
2 x − y ≥ −1
i. (1, −2)
2 (1) − 3 ( −2 ) ≤ 9
2 (1) − ( −2 ) ≥ −1
2+6 ≤ 9
2 + 2 ≥ −1
8 ≤ 9 true
4 ≥ −1 true
(1,-2) is a
solution.
ii. (−1, 2)
2 ( −1) − 3 ( 2 ) ≤ 9
2 ( −1) − ( 2 ) ≥ −1
−2 − 6 ≤ 9
−2 − 2 ≥ −1
−8 ≤ 9 true
−4 ≥ −1 false
(-1,2) is not
a solution.
iii. (3, −1)
2 ( 3 ) − 3 ( −1) ≤ 9
2 ( 3 ) − ( −1) ≥ −1
6+4≤9
6 + 1 ≥ −1
10 ≤ 9 false
7 ≥ −1 true
(3,-1) is not
a solution.
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Topic 7.6,
1.1, Slide 15
Graphing a System of Linear
Inequalities in Two Variables
Steps for Graphing Systems of Linear Inequalities in
Two Variables
Step 1. Use the Steps for Graphing Linear Inequalities in Two
Variables to graph each inequality.
Step 2. Determine where the shaded regions overlap, if any. This
overlapped (or shared) region represents the set of all
solutions to the system of inequalities.
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Topic 7.6,
1.1, Slide 16
Graphing a System of Linear
Inequalities in Two Variables
EXAMPLE
Graph each system of linear inequalities in two variables.
x+ y >2
a.
2x − y ≤ 6
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Topic 7.6,
1.1, Slide 17
Graphing a System of Linear
Inequalities in Two Variables
EXAMPLE
Graph each system of linear inequalities in two variables.
x −3y > 6
b.
2 x − 6 y < −9
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Topic 7.6,
1.1, Slide 18
Graphing a System of Nonlinear
Inequalities in Two Variables
Steps for Graphing Systems of Linear Inequalities in
Two Variables
Step 1. Use the Steps for Graphing Nonlinear Inequalities in
Two Variables to graph each inequality.
Step 2. Determine where the shaded regions overlap, if any. This
overlapped (or shared) region represents the set of all
solutions to the system of inequalities.
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2012,
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Topic 7.6,
1.1, Slide 19
Graphing a System of Nonlinear
Inequalities in Two Variables
EXAMPLE
Graph each system of nonlinear inequalities in two variables.
x 2 + y 2 ≤ 25
a.
x − y >1
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Topic 7.6,
1.1, Slide 20
Graphing a System of Nonlinear
Inequalities in Two Variables
EXAMPLE
Graph each system of nonlinear inequalities in two variables.
x − y2 ≥ 4
b.
x− y≤6
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Topic 7.6,
1.1, Slide 21