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

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