1. Solving and Graphing Linear Inequalities
Chapter 5
Solving and Graphing Linear Inequalities

2. 5.1 Solve Inequalities with Addition and Subtraction.
I can solve inequalities using addition and subtraction.
CC.9-12.A.REI.3

3. Graphing Inequalities on a Number Line
** the graph shows all possible solutions of the inequality

4. Solving an Inequality:
Solving a linear inequality in one variable is much like solving a linear equation in one variable.
Isolate the variable on one side using inverse operations.
Solve using addition:
Add the same number to EACH side.

5. Solving Using Subtraction:
Subtract the same number from EACH side.

6. Using Subtraction:
Graph the solution.

7. Using Addition:
Graph the solution.

8. Homework:
p. 301 #1, 2, 3-27odd, 29, 32, 36

9. 5.2 Solve Inequalities Using Multiplication and Division
I can solve inequalities using multiplication and division.
CC.9-12.A.REI.3

10. Solving Inequalities Using Multiplication:
Multiply each side by the same positive number.

11. Solving Inequalities Using Division:
Divide each side by the same positive number.

12. Multiply or Divide by a Negative Number:
NOTE:
When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

13. Solving by multiplication of a negative #
Multiply each side by the same negative number and REVERSE the inequality symbol.

14. Solving by dividing by a negative #
Divide each side by the same negative number and reverse the inequality symbol.

15. Example: You are shopping for bicycles
Example: You are shopping for bicycles. The type you want costs at least $185. You have saved $97. Find the possible amounts of money you need to save to buy the bicycle you want.
LSowatsky
This Photo by Unknown Author is licensed under CC BY-SA

16. Homework:
p. 308 #1, 2, 3-35 odd, 36, 37, 39

17. 5.3 Solve Multi-Step Inequalities
I can solve multi-step inequalities.
CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.A.REI.3

18. Multi-Step Inequalities:
Contains more than one operation
Use inverse operations to undo each operation in reverse order
Don’t forget to reverse the inequality symbol when multiplying or dividing by a negative number.
LSowatsky

19. Example: Solve 5m - 8 > 12 Graph the solution.

20. Example: Solve 12 - 3a > 18 Graph the solution.

21. Example: Solve the inequality. Graph the solution.

22. Example: Solve the inequality. Graph the solution.

23. Homework:
p. 314 #1, 2, 3 – 31 odd, 34, 35,

24. 5.4 Solve Compound Inequalities
I can solve compound inequalities.
CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.A.REI.3

25. Difference between and and or
AND means intersection
-what do the two items have in common?
OR means union
-if it is in one item, it is in the solution A B A B

26. ● ● Example: Graph x < 4 and x ≥ 2 a) Graph x < 4 o o3 4 2 o 3 4 2 o
b) Graph x ≥ 2 3 4 2 ● ●
c) Combine the graphs
d) Where do they intersect?

27. ● ● Example: Graph x < 2 or x ≥ 4 a) Graph x < 2 o o3 4 2 o 3 4 2 o
b) Graph x ≥ 4 3 4 2 ● 3 4 2 ●
c) Combine the graphs

28. 3) Which inequalities describe the following graph?-2 -1 -3 o
y > -3 or y < -1
y > -3 and y < -1
y ≤ -3 or y ≥ -1
y ≥ -3 and y ≤ -1

29. Example: Solve and graph the solution. 2x + 3 < 9 or 3x – 6 > 12
LSowatsky

30. Example: Graph 3 < 2m – 1 < 9

31. Homework:
p. 326 #1, 2, 3 – 35 odd, 38, 40, 44

32. 5.5 Solve Absolute Value Equations
I can solve absolute value equations
CC.9-12.A.CED.1, CC.9-12.A.CED.3, CC.9-12.F.IF.7b

33. Absolute Value: Definition – distance from the origin Ex: |3| = 3
|-5| = 5

34. Absolute Value Equations
If |x| = 3, what do you know about x?
Remember: Absolute Value is a distance.
x has a distance of 3 from zero.
If x is 3 ‘steps’ from zero on the number line, what could the value of x be?

35. Solving an Absolute Value Equation
The equation is equivalent to the statement ax + b = c or ax + b = -c

36. Example: Solve

37. Example: Solve

38. Example: Solve, if possible.

39. Absolute Deviation:
The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value.
Absolute deviation =

40. Example: A volleyball league is preparing a two minute radio ad to announce tryouts. The ad has an absolute deviation of 0.05 minute. Find the minimum and maximum acceptable times the radio ad can run.

41. Homework:
p. 335 #1, 2, 3–35odd, 38, 39, 42, 43,

42. Graph Absolute Value Functions (time permitting):
This Photo by Unknown Author is licensed under CC BY-SA

43. An absolute value function is a piecewise function that consists of two rays and is V-shaped. The corner point (turning point) of the graph is the vertex.
This Photo by Unknown Author is licensed under CC BY

44. To Graph:
To graph an absolute value function, it is useful to make a table of values.
You can also use symmetry to get more values.
Be sure to graph the vertex.

45. Parent Graph:

46. Graph and compare with the parent function.

47. Graph and compare to the parent function.

48. Graph and compare to the parent function.

49. Homework:
p. 339 #1-7

50. 5.6 Solve Absolute Value Inequalities
I can solve absolute value inequalities.
CC.9-12.A.CED.1, CC.9-12.A.CED.3

51. What does really mean?
All the numbers whose distance from zero is greater than 4. -4 4 or
On this slide, convince students about the less than -4 piece by asking them how far -5, -6, -20 etc. are from zero. Then ask if that distance is greater than 4 from zero.
*Notice that you need to have two inequalities to represent the distances that are greater than 4 from zero.
© 2011 The Enlightened Elephant

52. What does really mean? -4 4 and
All the numbers whose distance from zero is less than 4. -4 4
and
*Notice that you need to have two inequalities to represent the distances that are less than 4 from zero.
On this slide, convince students about the greater than -4 piece by asking them how far -3, -2 are from zero. Then ask if that distance is less than 4 from zero.
However, since these inequalities must happen at the same time, it should be written as
© 2011 The Enlightened Elephant

53. Absolute Value Inequalities Rules:If
AND
-c < ax + b < c OR
ax + b < -c or ax + b > c

54. You Try! Solve and graph. or Final answer: -3 4
© 2011 The Enlightened Elephant

55. You Try! Solve and graph. Final answer: -2 6
© 2011 The Enlightened Elephant

56. You Try! Solve and graph. Final answer: -3 2
© 2011 The Enlightened Elephant

57. You Try! Solve and graph. Final answer: NO SOLUTION!
Absolute values are distances, which cannot be negative.
You can choose any value for x and the left side of the inequality will always be positive.
Positive numbers are NEVER less than negative numbers.
© 2011 The Enlightened Elephant

58. Homework:
p. 343 #1, 2, 3 – 31 odd, 35, 36

59. 5.7 Graph Linear Inequalities in Two Variables
I can graph linear inequalities in two variables.
CC.9-12.A.CED.3, CC.9-12.A.REI.12

60. Inequalities in Two Variables:
The solution of an inequality in two variables x and y is an ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality.
The graph is the set of points that shows all solutions of the inequality.

61. Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a
Check whether the ordered pairs are solutions of 2x - 3y ≥ -2. a. (0, 0) b. (0, 1) c. (2, -1)

62. 2x – 3y ≥ -2 What it looks like:
Every point in the shaded region is a solution of the inequality and every other point is not a solution. 3 2 1 -1 -2 -3
2x – 3y ≥ -2

63. Steps to graphing a linear inequality:
Sketch the graph of the corresponding linear equation.
Use a dashed line for inequalities with < or >.
Use a solid line for inequalities with ≤ or ≥.
This separates the coordinate plane into two half-planes.
Half-plane: created by the boundary line; only one half-plane contains the solutions

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

65. Sketch the graph of y – 2x < 4

66. Sketch the graph of 2x – y ≥ 1

67. Sketch the graph of x + 8y >16

68. Sketch the graph y < 6.

69. Homework:
p. 351 # 1, 2, 3 – 51 odd, 53, 54

70. Review:
P.361 #17 – 34 (chapter test from book)

71. Chapter Test