1. Graphing a Linear Inequality Graphing a linear inequality is very similar to graphing a linear equation.

2. Graphing a Linear Inequality 1) Solve the inequality for y (or for x if there is no y). 2) Change the inequality to an equation and graph. 3) If the inequality is, the line is dotted. If the inequality is ≤ or ≥, the line is solid.

3. Linear Inequalities A linear inequality in two variables can be written in any one of these forms:  Ax + By < C  Ax + By > C  Ax + By ≤ C  Ax + By ≥ C An ordered pair (x, y) is a solution of the linear inequality if the inequality is TRUE when x and y are substituted into the inequality.

4. Example 1 Which ordered pair is a solution of 5x - 2y ≤ 6? A.(0, -3) B.(5, 5) C.(1, -2) D.(3, 3)

5. Graphing Linear Inequalities The graph of a linear inequality is the set of all points in a coordinate plane that represent solutions of the inequality. – We represent the boundary line of the inequality by drawing the function represented in the inequality.

6. Graphing Linear Inequalities The boundary line will be a: – Solid line when ≤ and ≥ are used. – Dashed line when are used. shaded Our graph will be shaded on one side of the boundary line to show where the solutions of the inequality are located.

7. Graphing Linear Inequalities Here are some steps to help graph linear inequalities: 1.Graph the boundary line for the inequality. Remember:  ≤ and ≥ will use a solid curve.  will use a dashed curve. 2.Test a point NOT on the boundary line to determine which side of the line includes the solutions. (The origin is always an easy point to test, but make sure your line does not pass through the origin)  If your test point is a solution (makes a TRUE statement), shade THAT side of the boundary line.  If your test points is NOT a solution (makes a FALSE statement), shade the opposite side of the boundary line.

8. Example 2 Graph the inequality x ≤ 4 in a coordinate plane. HINT: Remember HOY VEX. Decide whether to use a solid or dashed line. Use (0, 0) as a test point. Shade where the solutions will be. y x 5 5 -5

9. Graphing a Linear Inequality Graph the inequality 3 - x > 0 First, solve the inequality for x. 3 - x > 0 -x > -3 x < 3

10. Graph: x<3 Graph the line x = 3. Because x < 3 and not x ≤ 3, the line will be dotted. Now shade the side of the line where x < 3 (to the left of the line). 6 4 2 3

11. Graphing a Linear Inequality 4) To check that the shading is correct, pick a point in the area and plug it into the inequality. 5) If the inequality statement is true, the shading is correct. If the inequality statement is false, the shading is incorrect.

12. Graphing a Linear Inequality Pick a point, (1,2), in the shaded area. Substitute into the original inequality 3 – x > 0 3 – 1 > 0 2 > 0 True! The inequality has been graphed correctly. 6 4 2 3

13. Example 3 Graph 3x - 4y > 12 in a coordinate plane. Sketch the boundary line of the graph.  Find the x- and y-intercepts and plot them. Solid or dashed line? Use (0, 0) as a test point. Shade where the solutions are. y x 5 5 -5

14. Example 4: Using a new Test Point Graph y < 2 / 5 x in a coordinate plane. Sketch the boundary line of the graph.  Find the x- and y-intercept and plot them.  Both are the origin! Use the line’s slope to graph another point. Solid or dashed line? Use a test point OTHER than the origin. Shade where the solutions are. y x 5 5 -5

15. Absolute Value Functions and Graphs Graphing Absolute Value Functions

16. Absolute Value Functions An absolute value function is a function with an absolute value as part of the equation… f(x) = |mx + b| Graphs of absolute value equations have two special properties: a) a vertex b) they look like angles

17. Absolute Value Functions Vertex – point where the graph changes direction

18. Finding the Vertex For an equation y = |mx + b| + c, vertex = -b, c m Example:Find the vertex of y = |4x + 2| - 3

19. Finding the Vertex For an equation y = |mx + b| + c, vertex = -b, c m Example:Find the vertex of y = |4x + 2| - 3 Answer:= -2, -3 4 = -1, -3 2

20. Absolute Value Functions Steps to graphing an absolute value function… 1.Find the vertex 2.Write two linear equations and find slope 3.Use slope to plot points, connect the dots

21. Absolute Value Functions Example 1: Graph y = |3x + 12|.

22. Absolute Value Functions Step 1: Find the vertex y = |3x + 12| m = -b = c =

23. Absolute Value Functions Step 1: Find the vertex y = |3x + 12| m = 3 -b = -12 c = 0

24. Absolute Value Functions Step 1: Find the vertex y = |3x + 12| m = 3 -b = -12 c = 0 vertex=-b, c m vertex=-12, 0 3 vertex=(-4, 0)

25. Absolute Value Functions Step 1: Find the vertex vertex = (-4, 0)

26. Absolute Value Functions Step 2: Write two linear equations and find slope. y = |3x + 12| PositiveNegative

27. Absolute Value Functions Step 2: Write two linear equations and find slope. y = |3x + 12| PositiveNegative y = 3x + 12 y = -3x – 12 m 1 = m 2 =

28. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-4, 0) m 1 = 3 m 2 = -3

29. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-4, 0) m 1 = 3 m 2 = -3

30. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-4, 0) m 1 = 3 m 2 = -3

31. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-4, 0) m 1 = 3 m 2 = -3

32. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-4, 0) m 1 = 3 m 2 = -3

33. Absolute Value Functions Example 2: Graph y = |3x + 6| - 2

34. Absolute Value Functions Step 1: Find the vertex y = |3x + 6| - 2

35. Absolute Value Functions Step 1: Find the vertex y = |3x + 6| - 2 m = 3 -b = -6 c = -2 vertex = (-6/3, -2) = (-2, -2)

36. Absolute Value Functions Step 1: Find the vertex vertex = (-2, -2)

37. Absolute Value Functions Step 2:Write two linear equations and find slope y = |3x + 6| - 2 PositiveNegative

38. Absolute Value Functions Step 2:Write two linear equations and find slope y = |3x + 6| - 2 Positive Negative y + 2 = 3x + 6 y + 2 = -3x – 6 y = 3x + 4 y = -3x – 8

39. Absolute Value Functions Step 2:Write two linear equations and find slope y = |3x + 6| - 2 Positive Negative y + 2 = 3x + 6 y + 2 = -3x – 6 y = 3x + 4 y = -3x – 8 m= 3m= -3

40. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3

41. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3

42. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3

43. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3

44. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3

45. Absolute Value Functions Step 3: Use the slope to plot points vertex = (-2, -2) m 1 = -3 m 2 = 3