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Do NowJournal:

“A person must be at least 17 years old to get a New Jersey driver’s license.” What would the graph of that inequality look like? Why would we want to graph the solution? Why not just list the solutions?

Solution of Linear Inequalities

Graphing Linear Inequalities in Two Variables

January 2, 2014

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Solution of Linear Inequalities

Solve and graph the following inequalities.

1.2(m-3) + 7 < 21

2.4(n – 2) – 6 > 18

3.-6(2x – 10)+ 12x ≤ 180

<

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The solution set of a linear inequality in two variables is the set of all points that satisfy the inequality. It forms a region – the line may or may not be part of the solution set.

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.

Example: The solution set for x + 2y ≤ 8 is the shaded region. x

y

2

2

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x

y

x

yIf the inequality is < or >, the boundary line is dotted; its points are not solutions.

If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.

Example: The boundary line of the solution set of x + y < 2 is dotted.

Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.

3x – y < 2

3x – y = 2

3x – y > 2

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x

y

Example: For 2x – 3y ≤ 18 graph the boundary line.

The solution set is a half-plane.

A test point can be selected to determine which side of the half-plane to shade.

Shade above and to the left of the line.

Use (0, 0) as a test point.

(0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions.

(0, 0)

2-2

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To graph the solution set for a linear inequality:

2. Select a test point, not on the boundary line, and determine if it is a solution.

3. Shade a half-plane.

1. Graph the boundary line.

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x

y

Example: Graph the solution set for x – y > 2.

1. Graph the boundary line x – y = 2 as a dotted line.

2. Select a test point not on the line, say (0, 0).

(0) – 0 = 0 > 2 is false.

3. Since this is a not a solution, shade in the half-plane not containing (0, 0).

(0, 0)

(2, 0)

(0, -2)

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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.

x

y

4

4

- 4

- 4

x

y

4

4

- 4

- 4

Example: Graph the solution set for x ≥ 4.