6 5 Graphing Linear Inequalities in Two Variables

6. 5 Graphing Linear Inequalities in Two Variables Wow, graphing really is fun!

What is a linear inequality? • A linear inequality in x and y is an inequality that can be written in one of the following forms. • ax + by < c • ax + by ≤ c • ax + by > c • ax + by ≥ c

• An ordered pair (a, b) is a solution of a linear equation in x and y if the inequality is TRUE when a and b are substituted for x and y, respectively. • For example: is (1, 3) a solution of 4 x – y < 2? • 4(1) – 3 < 2 • 1 < 2 This is a true statement so (1, 3) is a solution.

Check whether the ordered pairs are solutions of 2 x - 3 y ≥ -2. a. (0, 0) b. (0, 1) c. (2, -1) (x, y) Substitute A (0, 0) 2(0) – 3(0) B (0, 1) 2(0) – 3(1) C (2, -1) 2(2) – 3(-1) Conclusion = 0 ≥ -2 (0, 0) is a solution. = -3 ≥-2 (0, 1) is NOT a solution. = 7 ≥ -2 (2, -1) is a solution.

Graph the inequality 2 x – 3 y ≥ -2 3 2 1 -3 -2 -1 -1 -2 -3 1 2 3 4 Every point in the shaded region is a solution of the inequality and every other point is not a solution.

Steps to graphing a linear inequality: 1. Sketch the graph of the corresponding linear equation. 1. Use a dashed line for inequalities with < or >. 2. Use a solid line for inequalities with ≤ or ≥. 3. This separates the coordinate plane into two half planes.

2. Test a point in one of the half planes to find whether it is a solution of the inequality. 3. If the test point is a solution, shade its half plane. If not shade the other half plane.

Sketch the graph of 6 x + 5 y ≥ 30 1. Write in slopeintercept form: y ≥ - 6/ 5 x + 6 This will be a solid line. 2. Test a point. (0, 0) 6(0) + 5(0) ≥ 30 0 ≥ 30 Not a solution. 3. Shade the side that doesn’t include (0, 0). 6 4 2 -6 -4 -2 -2 -4 -6 2 4 6 8

Sketch the graph y < 6. 1. This will be a dashed line at y = 6. 2. Test a point. (0, 0) 0 < 6 This is a solution. 3. Shade the side that includes (0, 0). 6 4 2 -6 -4 -2 -2 -4 -6 2 4 6 8

Sketch the graph of 2 x – y ≥ 1 1. Write in slopeintercept form: y = 2 x – 1 This will be a solid line. 2. Test a point. (0, 0) 2(0) - 0 ≥ 1 Not a solution. 3. Shade the side that doesn’t include (0, 0). 3 2 1 -3 -2 -1 -1 -2 -3 1 2 3 4