Graphing Linear Inequalities in Two Variables

Section 6.5

Algebra I

Definitions

Linear inequality: A linear inequality in x and y is an inequality that can be written as follows

Solution: An ordered pair (x,y) is a solution of a linear inequality if the inequality is true when the values of x and y are substituted into the inequality

ax by c

ax by c ax by c ax by c

Example 1

Check whether the ordered pair is a solution of 2x-3y>-2 (0,0)

(0,1)

(2,-1)

Example 1 Continued

For (0,0) both x and y are 0. Substitute 0 for x and 0 for y

2(0)-3(0) > -2 0-0 > -2 0 > -2

Since 0 is greater than -2, then (0,0) is a solution to the inequality

Example 1 continued

To check if (0,1) is a solution, we use x=0 and y=1 2(0)-3(1) > -2 0-3 > -2 -3>-2

Since -3 is not greater than -2, then (0,1) is not a solution

Example 1 continued

You check if (2,-1) is a solution

Definitions

Graph: The graph of a linear inequality in two variables is the graph of the solutions of the inequality

Half-plane: In a coordinate plane, the region on either side of a boundary line.

Example 2

Sketch a graph of y>-3 To do this, we will expand on our graphs from

before. We are going to use a coordinate plane rather than a number line

Use a dotted line for less than or greater than Use a solid line for less than or equal to and

greater than or equal to

Example 2 continued Start by graphing y=-3

Example 2 continued Next, we need to shade in all values where

y>-3

Example 3

Try to sketch the graph of x≤5. Remember, start with x=5. Is this a solid or dotted line? Then shade in where x≤5.

Example 3 continued

Did you get this?

Example 4

If you are given x-3>5…. How would you graph this?

First, solve for x. x>8 Then graph as we did in the previous two

examples.

Example 5

Sketch the graph of x – y < 2. First, graph the line x – y = 2. We would solve for y…

-y = -x + 2 y = x – 2.

Now, we know the slope is one and the y-intercept is -2.

We also will graph using a dotted line.

Example 5 Continued

Example 5 continued

To decide where to share, we can test a point on one side of the line. I like to test (0,0) if possible.

For x – y < 2….. We can use x = 0, y = 0. 0 – 0 < 2 0 < 2

Since this is a true statement, we shade where (0,0) is at on the graph – as well as that side of the graph.

Example 5 Continued