Chapter 3 Graphs and Functions 3.3 – Linear Functions: Graphs and Applications 3.4 – The Slope-Intercept Form of a Linear Equation 3.5 – The Point-Slope Form of a Linear Equation 3.6 – The Algebra of Functions 3.7 – Graphing Linear Inequalities

3.7 Graphing Linear Inequalities Objectives: 1. Determine whether an ordered pair is a solution for a linear inequality with two variables. 2. Graph linear inequalities.

Graph Linear Inequalities Linear Inequality in Two Variables A linear inequality in two variables can be written in one of the following forms: ax + by < c, ax + by > c, ax + by ≤ c, ax + by ≥ c where a, b, and c are real numbers and a and b are not both 0. Examples: 2x + 3y > 2 -x – 2y ≤ 3 3y < 4x - 9

Example 1: Determine whether (3, –5) is a solution for y  3x + 4. – 5  3(3) + 4 –5  13 This statement is true, so (3, –5) is a solution. Replace x with 3 and y with 5. True or False?

Graph Linear Inequalities Consider the graph of the equation x + y = 3. The line acts as a boundary between two half-planes and divides the plane into three distinct sets of points.

Graph Linear Inequalities To get the equation of the boundary line, replace the inequality symbol with an equals sign. Draw the graph of the equation in step 1. If the original inequality contains a  or  symbol, draw the boundary line using a solid line. If the original inequality contains a < or > symbol, draw the boundary line using a dashed line. Select any point not on the boundary line and determine if this point is a solution to the original inequality. If the point selected is a solution, shade the half-plane on the side of the line containing this point. If the selected point does not satisfy the inequality, shade the half-plane on the side of the line not containing the point.

Checkpoint (0, 0 ) satisfies the inequality. Example 2: Graph y < 2x + 1. Boundary Line: y = 2x + 1 dashed line Intercepts (0,1) (-1/2, 0) Test Point (0, 0) 0 < 2(0)+1 True Checkpoint (0, 0 ) satisfies the inequality.

Example 3: Graph: y > 4x + 1 Solution Graph the related equation: y = 4x + 1. (0, 1), (1/4, 0) and (1, 3) Select a test point: (0, 0) y > 4x + 1 0 > 4(0) + 1 0 > 1 False Shade other side of the boundary line.

Example 4: Graph: x + 4y ≥ 8 Solution Graph the related equation: x + 4y = −8 (0, 2) and (−8, 0) Select a test point: (0, 0) x – 4y ≥ 8 0 – 4(0) ≥ 8 0 ≥ 8 True Shade the region that contains the point.

Example 5: Graph: x > 4 Solution First graph the related line x = 4, using a dashed line. Select a test point: (0, 0) x > 0 0 > 0 False Shade other side of the boundary line.

Example 6: Graph: y  −1 Solution First graph the related line y = −1, using a solid line. Select a test point: (0, 0) 0  −1 False Shade other side of the boundary line.