Graphing Inequalities in Two Variables

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Graphing Inequalities in Two Variables
LP-L2 Objectives Graph a linear inequality in
two variables on a Cartesian Plane
Learning Outcome B-1
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• When you place a line on a coordinate plane, the
  line creates three regions
• the region that is the line itself
• the region to the left of the line
• the region to the right of the line
• Consider the graph of the equation y 2x
  1shown. It separates the coordinate plane into
  two half-planes. One half-plane, Region 1, is on
  the left of the line. The point A with
  coordinates (-4, 3) is in Region 1.

Theory Inequalities in Two Dimensions
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If the x-coordinate of point A, -4, is
substituted into the equation of the line to find
the corresponding y-coordinate on the line, the
result is y 2x 1y 2(-4) 1, ory -7.
In other words, the y-value for the point A is
greater than the y-value for the line. Use
another point in Region 1 to verify the
relationship. The y- values in Region 1 for a
given x-value are greater than the y-values on
the line.This region or half-plane can be
represented by the inequality y gt 2x 1.
Theory Inequalities in Two Dimensions
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What inequality do you think represents the area
in Region 2? The other half-plane (Region 2) is
on the right ofthe line. The point B with
coordinates (2, 2) is in Region 2.If the
x-coordinate of point B, 2, is substituted
intothe equation of the line to find the
correspondingy-coordinate on the line, the
result isy 2x 1y 2(2) 1, ory 5.
In other words, the y-value for the point B is
less than the y-value for the line. The y- values
in Region 2 for a given x-value are less than the
y-values on the line. This region or half-plane
can be represented by the inequality y lt 2x 1.
Theory Inequalities in Two Dimensions
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SummaryA line divides the coordinate plane into
two half-planes, which can each be represented by
an inequality statement.In our example, the
line y 2x 1 divides the coordinate plane into
two halves.Given the lines positive slope, the
left half can be represented by the equation y gt
2x 1, and the right half can be represented by
the equation y lt 2x 1.
Theory Inequalities in Two Dimensions
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• ExampleGraph the region 2x y lt 4.
• Steps
• Solve for y.
• Determine dashed or solid line.
• Select a point and test in inequality.
• Shade accordingly. (If test was true, shade
  region that includes point. If false, shade
  other side of boundary.)

Example Inequalities in Two Dimensions
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ExampleGraph the region 2x y lt 4. Solution
Example Inequalities in Two Dimensions
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Graph the region
Test Yourself
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Graph the region
Test Yourself
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Graph the region
Test Yourself
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Graph the region
Test Yourself
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Graph the region
Test Yourself
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Special Case Graph the region
Test Yourself
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Special Case Graph the region
Test Yourself