6. 5 Graphing Linear Inequalities in Two Variables 7 6.5 Graphing Linear Inequalities in Two Variables 7.6 Graphing Linear Inequalities System Objectives 1. Graph linear inequalities. 2. Graph systems of linear inequalities.

Linear Inequalities A linear inequality in two variables is an inequality that can be written in the form Ax + By < C, where A, B, and C are real numbers and A and B are not both zero. The symbol < may be replaced with , >, or . The solution set of an inequality is the set of all ordered pairs that make it true. The graph of an inequality represents its solution set.

To Graph a Linear Inequality Step (1) Solve for y, convert the inequalities to Slope-Intercept Form. If one variable is missing, solve it and go to step (2). (2) Graph the related equation. ** If the inequality symbol is < or >, draw the line dashed. ** If the inequality symbol is  or , draw the line solid. (3) If y < or y  , shade the region BELOW the line If y > or y  , shade the region ABOVE the line If x < or x  , shade the region LEFT to the line If x > or x  , shade the region RIGHT to the line

Example Graph y > x  4. Already in S-I form. The related equation is y = x  4. Use a dashed line because the inequality symbol is >. This indicates that the line itself is not in the solution set. (3) Determine which half-plane satisfies the inequality. y > x  4 “>” shade above

Example Graph: 4x + 2y  8 1. Convert to S-I form: y  -2x + 4 2. Graph the related equation. 3. Determine which half-plane satisfies the inequality. “y ” shade below

Example Graph 2x – 4 > 0 . 1. One variable is missing. Solve it: 2. Graph the related equation x = 2 3. Determine which half-plane satisfies the inequality. “x >” shade the right.

Example Graph 8 - 3y  2 . 1. One variable is missing. Solve it: y  2 2. Graph the related equation y = 2 3. Determine the region to be shaded. “y ” shaded below

7.6 Systems of Linear Inequalities Graph the solution set of the system. First, we graph x + y  3 using a solid line. y  - x + 3 “above” Next, we graph x  y > 1 using a dashed line. y < x – 1 “below” The solution set of the system of equations is the region shaded both red and green, including part of the line x + y = 3.

Example Graph the following system of inequalities and find the coordinates of any vertices formed: We graph the related equations using solid lines. We shade the region common to all three solution sets.

Example continued The system of equations from inequalities (1) and (3): y + 2 = 0 x + y = 0 The vertex is (2, 2). The system of equations from inequalities (2) and (3): x + y = 2 The vertex is (1, 1). To find the vertices, we solve three systems of equations. The system of equations from inequalities (1) and (2): y + 2 = 0 x + y = 2 The vertex is (4, 2).

Summary To graph a two-variable linear inequality, graph the related equation first (variable y must be solved) with appropriate boundary line: “<” and “>” use dash line “≤” and “≥” use solid line “y < …” and “y ≤ …” shade the region below “y > …” and “y ≥ …” shade the region above “x < …” and “x ≤ …” shade the region left “x > …” and “x ≥ …” shade the region right When graphing the linear inequality system, follow the step 1 ~ 3 and choose the region shaded most.

Assignment 6.5 P 363 #’s 10 - 22 (even), 32 - 40 (even), 45 - 60 (even) 7.6 P 435 #’s 9 - 26