Linear Inequalities

Honors Math – Grade 8

Graphing Linear Inequalities in Two Variables The solution set for an inequality in two

variables contains many ordered pairs when the domain and range are the set of real numbers.

Any line in the plane divides the plane into two regions called half planes. The line is called the boundary of each of the two half-planes.

Key Concept

Half Plane

Half Plane

Boundary Line

Graphing the boundary line is the first step in graphing an inequalities solution.

4y4y

x y

0 4

1 4

2 4

1. Determine the equation of the boundary line by replacing the inequality sign with an equals sign.

2. Graph the equation of the boundary line.

Dashed Line - used to graph > or < inequalities because the boundary is not part of the solution set.

4y

3. Choose a point in each half plane.

4. Shade the entire half-plane whose point gave a true statement.

Test each point in the inequality.

32

1 xy3

2

1 xy

1. Determine the equation of the boundary line by replacing the inequality sign with an equals sign.

2. Graph the equation of the boundary line.

Dotted Line because inequality is >.

32

1 xy

3. Choose a point in each half plane.

Since (0, 0) gives a false statement, shade the other half-plane.

Test each point in the inequality.

m = 1/2y-int = 3

Use the origin as a standard test point.

42 xy42 xy

1. Determine the equation of the boundary line by replacing the inequality sign with an equals sign.

2. Graph the equation of the boundary line. Solve for y.

Solid Line - used to graph > or < inequalities because the boundary is part of the solution set.

42 xy

3. Choose a point in each half plane.

Since (0, 0) gives a false statement, shade the other half-plane.

Test each point in the inequality.

Use the origin as a standard test point.

1x1x

x y

-1 4

-1 4

-1 4

1. Determine the equation of the boundary line by replacing the inequality sign with an equals sign.

2. Graph the equation of the boundary line.

Solid Line - used to graph > or < inequalities because the boundary is part of the solution set.

1x

3. Choose a point in each half plane.

4. Shade the entire half-plane whose point gave a true statement.

Test each point in the inequality.

624 yx624 yx

1. Determine the equation of the boundary line by replacing the inequality sign with an equals sign.

2. Graph the equation of the boundary line. Standard Form.

Dotted Line because inequality is >.

624 yx

3. Choose a point in each half plane.

Since (0, 0) gives a false statement, shade the other half-plane.

Test each point in the inequality.

Use the origin as a standard test point.

Define the variables

Lee Cooper writes and edits short articles for a local newspaper. It takes her about an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can she write and edit in one day?

Let x = the # of articles Lee can write and y = the # of articles Lee can edit. Write an open sentence representing this

situation.The # of articles she can write + ½ times the number she can edit is up to 8 hours.

82

1 yx 8

2

1 yx

1. Determine the equation of the boundary line.

2. Graph the equation of the boundary line. Solve for y.

Solid Line because inequality is <.

3. Choose a point in each half plane.

Use the origin as a standard test point.

Since (0, 0) gives a true statement, shade the half-plane but not negative integers.

One solution is Lee could write 2 articles and edit 4.

Define the variables

Tickets for the school play cost $5 for students and $7 for adults. The school wants to earn at least $6300 on each performance.

Let x = the # of student tickets and y = the # of adult tickets. Write an open sentence representing this

situation.5 times the # of student tickets + 7 times the number of adult tickets is at most $6300

630075 yx 630075 yx

1. Determine the equation of the boundary line.

2. Graph the equation of the boundary line. Solve for y.

Solid Line because inequality is <.

3. Choose a point in each half plane.

Use the origin as a standard test point.

Since (0, 0) gives a false statement, shade the other half-plane.

One solution is the school could sell 600 student tickets and 650 adult tickets.