Graphing Linear Inequalities in Two Variables Lesson 3.6 Core Focus on Linear Equations

Determine if each point is on the given line. 1.Is the point (2, 4) on the line y = −4x + 4? 2.Is the point (−1, 5) on the line 3x + 2y = 7? 3.Is the point (6, −2) on the line x = −2? Warm-Up No Yes No

Graphing Linear Inequalities in Two Variables Graph a linear inequality on a coordinate plane. Lesson 3.6

Step 1 Juniper is thinking of two numbers, x and y, whose sum is at least 6. a.Write an inequality that represents this statement. b.Find 6 pairs of numbers Juniper may be thinking of. List the sets of two numbers as ordered pairs (x, y). Step 2How many possible pairs of numbers fit Juniper’s description? Explain your reasoning. Step 3On a coordinate plane, graph at least three ordered pairs that have a sum of exactly six (remember that Juniper said the sum of her numbers is at least six). Draw a solid line through these points. This is the boundary line for the solution set. All values on this line have ordered pairs that have a sum of 6. Step 4Plot at least three of your ordered pairs from Step 1 on your graph. Are these numbers “above” or “below” your boundary line? Is it possible for some of your points to be on one side of the boundary line and other points on the opposite side? Explain your reasoning.

Step 5Shade the coordinate plane on the side of the boundary line you placed your ordered pairs on in Step 4. This shows all of the possible solutions to Juniper’s description of her two numbers. Choose another point in your shaded area as a test point. Do the x- and y-values add up to more than 6? If so, you shaded the correct side of the boundary line. Step 6On a new coordinate plane, begin to graph the inequality by graphing three points on the line. DO NOT connect the points with a solid line. Test one of the points on your line in your inequality. What do you notice? Step 7Connect your points with a dashed line to show that these points are not included in the solution. Choose a test point on either side of the boundary line. Substitute the point into the original inequality. If the x- and y-values make the statement true, shade that side of the boundary line. If they do not, shade the side that does not contain your test point. Step 8Graph y ≤ −2x + 3 on a new coordinate plane.

Graphing Linear Inequalities in Two Variables 1.Graph the boundary line by replacing the inequality sign with an equal sign. a.If the inequality sign is > or <, use a dashed boundary line. b.If the inequality sign is ≥ or ≤, use a solid boundary line. 2.Choose a test point which is not on the boundary line. Substitute the x- and y-values into the original inequality. 3.If the test point makes a true statement when substituted into the inequality, shade the side of the boundary line that contains that point. If the statement is false, shade the side of the boundary line that does not contain the test point.

Graph y > 3x − 4. Graph the boundary line by graphing y = 3x – 4. Use a dashed line to connect the points since the inequality symbol is >. Choose a test point that is not on the boundary line. (–2, 4) Substitute the coordinates into the original inequality to see if the statement is true or false. Four is greater than negative 10 so the statement is true. Shade the entire side of the boundary line that contains the test point (–2, 4). Example 1 ? The shaded region shows all the possible ordered pairs that are solutions to y > 3x – 4.

Graph x + 2y  4. Rewrite the inequality as an equation. Convert to slope-intercept form. Graph the boundary line by graphing y = x + 2. Use a solid line to connect the points since the inequality is . Choose a test point that is not on the boundary line.(4, 8) Example 2 x + 2y = 4 –x 2y = –x + 4 y = x + 2

Graph x + 2y  4. Substitute the coordinates into the original inequality to see if the statement is true or false. Twenty is NOT less than or equal to 4 so the statement is FALSE. Shade the entire side of the boundary line that DOES NOT contain the test point (4, 8). Example 2 Continued… ?

Boundary lines can be solid or dashed. What does each type of line tell you about the inequality? Communication Prompt

Graph each linear inequality. 1. y > 2x − 1 2. Exit Problems