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Graphing Linear Inequalities in Two Variables

• A linear inequality in two variables takes one of the following forms:

• The solution of a linear inequality is all ordered pairs (x, y) that satisfy the inequality.

0Ax By c

0Ax By c

0Ax By c 0Ax By c

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• Example 1

Consider the linear inequality …

12x y

The ordered pair (8, - 4) is not a solution:

?

8 12 4

?8 16False

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12x y

The ordered pair (8, 6) is a solution:

?8 12 6

?8 6True

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• To graph a linear inequality in two variables:

1) Put an equals sign in place of the inequality sign and graph the line.

• If the inequality symbol is < or >, graph a dashed line.

• If the symbol is ≤ or ≥, graph a solid line.

• The line divides the plane into two half-planes.

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2) Try a test point that is not on the line in the original inequality.

• If the point satisfies the inequality, then shade in the half-plane that contains the point.

• If the point does not satisfy the inequality, then shade in the half-plane that does not contain the point.

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• Example 2

Graph:

Create the equation.

3 4 12x y

3 4 12x y

Graph the line by the method of your choice:1. Solve for y and use the slope-intercept

method to graph.2. Find the points (0, y) and (x, 0) and graph

the line through them.

Since the < symbol is used, draw a dashed line.

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2

-2

-4

-6

5 10

Select a point not on the line.

0,0

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Check the point in the original inequality.

3 4 12x y

?

3 0 4 0 12

?0 12

True

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2

-2

-4

-6

5 10

Since the point satisfied the inequality, shade in the same side that the point is on.

0,0

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• Example 3

Graph:

Create the equation.

3x

3x

Since the ≤ symbol is used, draw a solid line.

Graph the line. Recall that the graph of x = c is a vertical line.

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Select a point not on the line.

4

2

-2

-5

0,0

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Note that once again we have selected the origin, because it is the easiest to evaluate in the inequality.

The only time we won’t choose the origin is when the line itself (dashed or solid) passes through the origin.

Any point not on the line can be used, but the origin is very convenient.

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Check the point in the original inequality.

3x

?

0 3

False

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Since the point did not satisfy the inequality, shade in the opposite side that the point is on.

0,04

2

-2

-5

0,0

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