Graphing Linear Inequalities in Two Variables (5-6) Objective: Graph linear inequalities on the coordinate plane. Solve inequalities by graphing.

Graph Linear Inequalities The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a boundary, which divides the coordinate plane into two half-planes. The boundary may or may not be included in the graph. When it is included, the solution is a closed half-plane. When not included, the solution is an open half-plane.

Graph Linear Inequalities Use the following steps to graph a linear inequality. Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >. Use a test point to determine which half-plane should be shaded. Shade the half-plane that contains the solution.

Example 1 Graph 2y – 4x > 6. 2y – 4x > 6 +4x +4x 2y > 4x + 6 Graph the line y = 2x + 3 using a dashed line. 2y – 4x > 6 +4x +4x 2y > 4x + 6 2 2 2 y > 2x + 3 Test Point: (0, 0) 0 > 2(0) + 3 0 > 3 False

Check Your Progress Choose the best answer for the following. Graph y – 3x < 2. y – 3x < 2 A. B. +3x +3x y < 3x + 2 Graph the line y = 3x + 2 using a dashed line. Test Point: (0, 0) C. D. 0 < 3(0) + 2 0 < 2 True

Example 2 Graph x + 4y ≥ 2. x + 4y ≥ 2 -x -x 4y ≥ -x + 2 4 4 4 Graph the line y = -¼x + ½ using a solid line. x + 4y ≥ 2 -x -x 4y ≥ -x + 2 4 4 4 y ≥ -¼x + ½ Test Point: (0, 0) 0 ≥ -¼(0) + ½ 0 ≥ ½ False

Check Your Progress Choose the best answer for the following. Graph x + 2y ≥ 6. x + 2y ≥ 6 A. B. -x -x 2y ≥ -x + 6 2 2 2 y ≥ -½x + 3 Graph the line y = -½x + 3 using a solid line. C. D. Test Point: (0, 0) 0 ≥ -½(0) + 3 0 ≥ 3 False

Example 3 We can use a coordinate plane to solve inequalities with one variable. Use a graph to solve 2x + 3 ≤ 7. 2x + 3 ≤ 7 -3 -3 2x ≤ 4 2 2 x ≤ 2

Check Your Progress Choose the best answer for the following. Use a graph to solve 5x – 3 > 17. x > 20 x > 3 x < -4 x > 4 5x – 3 > 17 +3 +3 5x > 20 5 5

Example 4 When using inequalities to solve real-world problems, the domain and the range are often restricted to nonnegative or whole numbers. Ranjan writes and edits short articles for a local newspaper. It takes him about an hour to write an article and about a half-hour to edit an article. If Ranjan works up to 8 hours a day, how many articles can he write and edit in one day? x + ½y ≤ 8 -x -x 2( ) ½y ≤ -x + 8 y ≤ -2x + 16 Test Point: (0, 0) 0 ≤ -2(0) + 16 Any point in the shaded area is a solution. 0 ≤ 16 True Sample Answer: (2, 4) He can write 2 and edit 4 articles.

Check Your Progress Choose the best answer for the following. You offer to go to the local deli and pick up sandwiches for lunch. You have $30 to spend. Chicken sandwiches cost $3.00 each and tuna sandwiches are $1.50 each. How many sandwiches can you purchase for $30? 11 chicken sandwiches, 1 tuna sandwiches 12 chicken sandwiches, 3 tuna sandwiches 3 chicken sandwiches, 15 tuna sandwiches 5 chicken sandwiches, 9 tuna sandwiches 3x + 1.50y ≤ 30 3(11) + 1.50(1) ≤ 30 34.5 ≤ 30 False 3(12) + 1.50(3) ≤ 30 40.5 ≤ 30 False 3(3) + 1.50(15) ≤ 30 31.5 ≤ 30 False 3(5) + 1.50(9) ≤ 30 28.5 ≤ 30 True