1. SOLVING TWO VARIABLE LINEAR INEQUALITIES
INCLUDING ABSOLUTE VALUE INEQUALITIES

2. Summary of Inequality Signs
>
>
Shade above the line
<
<
Shade below the line
Continuous line
Dashed line

3. Graphing Linear Inequalities
The graph of a linear inequality is a region of the coordinate plane that is bounded by a line. This region represents the SOLUTION to the inequality.

4. A linear inequality is an inequality in two variables whose graph is a region of the coordinate plane that is bounded by a line.

5. Graph the following inequality: x > 2
Boundary is: x = 2 We shaded at the right of the line because x is more than 2. The line is dashed because it is not equal or less than x, so the line which is the boundary is not included in the solution.

6. Graph the following inequality: y < 6
Boundary: y = 0x + 6 m= 0 y- intercept = (0,6) We shaded below the line because y is less than 6. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included in the solution.

7. Example

8. 1. The boundary line is dashed.

9. Substitute (0, 0) into the inequality to decide where to shade.
So the graph is shaded away from (0, 0).

11. Graph the following inequality: 4x + 2y < 10
Solve for y  y < -2x + 5 Boundary is: y = -2x + 5 m= -2 y- intercept = (0,5) We shaded below the line because y is less than the expression -2x + 5. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included.

12. Graph the following inequality: -9x + 3y< 3
Solve for y  y < 3x + 1 Boundary is: y = 3x + 1 m= 3 y- intercept = (0, 1) We shaded below the line because y is less than the expression 3x +1. The line is dashed because it is not equal or less than y, so the line which is the boundary is not included.

13. Graph the following inequality: y – 2 > (x – 4)
Solve for y  y > x – 3
Boundary is: y = x – m = y-intercept = (0, -3)
We shaded above because y is greater or equal than the expression
and the line is continuous because the word equal in greater or equal indicates
that the boundary is included in the solution.

14. EXAMPLE

15. EXAMPLE

16. EXAMPLE

17. EXAMPLE

18. Problem, con’t

19. Graph the following absolute value equation: y = |x|
For x < 0 For x > 0
y = -x y = x
Now let’s shift it two units up:
y = |x| + 2
Now let’s shift it three units to the right:
y = |x - 3| + 2
Now let’s graph it upside down
y = – |x-3| + 2
Now let’s make it skinner
y = – 6|x-3| + 2

20. So, that’s how the different parameters in an absolute value equation affect our graph. Now let’s graph absolute value inequalities.

21. Absolute Value Inequalities
Graph the absolute value function then shade above OR below
Solid line…y <, y>
Dashed line…y<, y>
Shade above y>, y>
Shade below…y<, y<

22. Absolute Value Inequalities
Graph y < |x – 2| + 3
y < |x – 2| + 3
DASHED line
Shade BELOW
slope = 1
Vertex = (2, 3)

23. Absolute Value Inequalities
Graph y < |x – 2| + 3
Vertex = (2, 3)

24. Absolute Value Inequalities
Graph y < |x – 2| + 3
slope = 1

25. Absolute Value Inequalities
Graph y < |x – 2| + 3
DASHED line
Shade BELOW

26. Absolute Value Inequalities
Graph y < |x – 2| + 3
Shade BELOW

27. Absolute Value Inequalities
Graph –y + 1 < -2|x + 2|
-y < -2|x + 2| - 1
y > 2|x + 2| + 1
-y so CHANGE the direction of the inequality

28. Absolute Value Inequalities
y > 2|x |
Solid line
Shade above
Vertex = (-2, 1)
Slope = 2

29. Absolute Value Inequalities
y > 2|x |

30. Absolute Value Inequalities
y > 2|x |

31. Absolute Value Inequalities
y > 2|x |

32. Absolute Value Inequalities
y > 2|x |

33. Absolute Value Inequalities
Write an equation for the graph below.

34. Graph the following inequality: y > |x|
Finding the boundary:
For x < 0 For x > 0
y = -x y = x .
There are two regions:
Testing point (0,2)
2 > | 0|
2 > true
Therefore, the region where (0,2) lies is the solution region and we shade it.

35. Graph the following inequality: y < |x+1|– 3
Finding the boundary: For x + 1 < 0 For x + 1 > 0 y = -(x+1) – 3 y = x+1 – 3 y = - x – 1 -3 y = - x – 4 y = x – 2 There are two regions: Testing point (0,0) 0 < | 0+1| – 3 0 < -2 false So the region where (0,0) lies is not in the solution region, therefore we shade the region below. .

36. Steps:
Decide if the boundary graph is solid or dashed.
Graph the absolute value function as the boundary.
Use the point (0, 0), if it is not on the boundary graph, to decide how to shade.

37. Graph y ≥ 2|x – 3| + 2

38. Graph y ≥ 2|x – 3| + 2
1. The boundary graph is solid.

39. y ≥ 2|x – 3| + 2
0 ≥ 2|0 – 3| + 2
0 ≥ 2|-3| + 2
0 ≥ 6 + 2
0 ≥ 8 False
So shade away from (0, 0).

42. The boundary graph is dashed.

45. Your Turn!
8. Graph
9. Graph

46. 8. 9.

47. Example 10

48. Example 10
y ≤ |x+4| y ≥ 2x+5