© William James Calhoun, 2001 OBJECTIVE: The student will graph inequalities in the coordinate plane. All of the inequalities we have dealt with in this chapter have been in one variable. Their graphs were on number lines with either open or closed points and running in one or both directions. Now we will look at inequalities with two variables. Just like when we looked at equations with two variables, we will leave number lines behind and look at coordinate planes. Remember, the x-axis is the horizontal axis and the y-axis is vertical. 7-8: Graphing Inequalities in Two Variables

© William James Calhoun, : Graphing Inequalities in Two Variables Remember how to graph y = 2x + 1. slopey-int The equation is in slope-intercept form. It is easiest to use the slope-intercept method of graphing. With inequalities, two things will change. On the next two slides, the changes are explained. ID the slope and intercept. Put a point at the y-intercept. Connect the dots with a line. The slope is which means rise 2 and run 1.

© William James Calhoun, : Graphing Inequalities in Two Variables Now look at the inequality: y > 2x + 1. Pick points from the coordinate plane. If the point makes the inequality true, put a red dot at that point. If the point makes the inequality false, put a blue dot at that point. (0, 0)0 > 0 + 1False PointCheck: ? (1, 1)1 > 2 + 1False(2, 1)1 > 4 + 1False(2, 0)0 > 4 + 1False(0, 1)1 > 0 + 1False(1, 0)1 > 0 + 1False(0, 2)2 > 0 + 1True(0, 3)3 > 0 + 1True(1, 3)3 > 2 + 1False(1, 2)2 > 2 + 1False(2, 2)2 > 4 + 1False(2, 3)3 > 4 + 1False(-1, 2)2 > True(-1, 1)-1 > True(-1, 0)0 > True(-1, -1)-1 > False(0, -1)3 > 0 + 1False(-1, -2)-2 > False(-2, 0)1 > True(-2, -1)-1 > True(-2, -2)-2 > True(-2, -3)-3 > False(-2, -4)-4 > False From this, you can see the pattern that is emerging. If we graphed all the points in-between these points, we would see the following. Would it surprise you that this is the graph of y = 2x + 1? But what is the deal with the dotted line? Notice only blue areas are on the line. All of the red area is true. All of the blue area is false. The next slide explains the new rules of graphing inequalities.

© William James Calhoun, : Graphing Inequalities in Two Variables All of the points that make y > 2x + 1 true are in the red-shaded area. The solution set of an inequality in two variables is a half-plane. The line which separates the coordinate plane into two half-planes is the line created when the inequality is turned into an equation and graphed. If the inequality has > or < in it, the line is not included in the solution set. This is where the dotted line comes from. If the inequality has “or equal to” in it, the line will be solid. Once you know one point on either side of the line is true, you know all the points on that side are true.

© William James Calhoun, : Graphing Inequalities in Two Variables The steps to graphing inequalities in two variables is: (1) Pretend the inequality symbol is an equal symbol and get the two points to make the line. Either by double-intercept (if in standard form) or by slope-intercept (if in other forms.) (2) If the inequality is, or  draw a dotted line - otherwise a solid line. (3) Pick a point that is not on the line. The best point is (0, 0). (4) Plug that point into the original inequality. (5A) If the point you plug in gives you a true answer, shade in all other points in the coordinate plane that are on the same side of the graphed line. (5B) If the point you plug in gives you a false answer, shade in all points in the coordinate plane that are on the opposite side of the graphed line.

© William James Calhoun, : Graphing Inequalities in Two Variables EXAMPLE 1: Graph y + 2x  3. Pretend the inequality is an equation and solve for y. y + 2x = 3 -2x y = -2x + 3 slopey-int Get the two points from the slope and y-intercept. Pause. Look at the original problem. It has “less than or equal to” in it. This means there should be a solid line drawn through the two points. Pick a point. (0, 0) is not on the line, so use it. Draw an “X” on the chosen point. Now determine if the chosen point makes the original inequality true. Plug in zero for x and zero for y. y + 2x  3 (0) + 2(0)   3 0  3 The chosen point makes the original problem true. That means all the points on the same side of the line are true. That whole side of the line must be colored in. Only two new steps: (1) Determining whether to use a solid or dotted line, and (2) Shading in one side of the line after testing a point.

© William James Calhoun, : Graphing Inequalities in Two Variables EXAMPLE 2: Graph 3x + y < 45. Pretend the inequality is an equation and find the intercepts. Plot the intercepts. (Each line will be five units.) Pause. Look at the original problem. It has plain “less than” in it. This means there should be a dotted line drawn through the two points. Pick a point. (0, 0) is not on the line, so use it. Draw an “X” on the chosen point. Now determine if the chosen point makes the original inequality true. Plug in zero for x and zero for y. 3x + y < 45 3(0) + (0) < < 45 The chosen point makes the original problem true. That means all the points on the same side of the line are true. That whole side of the line must be colored in. 3x + y = 45 3(0) + y = 45 y = 45 3x + y = 45 3x + (0) = 45 3x = 45 x = 15 0 < 45

© William James Calhoun, : Graphing Inequalities in Two Variables HOMEWORK Page 439 # odd