1. Notes for Algebra 1
Chapter 5

2. Solving Inequalities by Addition and Subtraction
5-1 Notes for Algebra 1
Solving Inequalities by Addition and Subtraction

3. Inequality
An open sentence that contains <, >, < , or >

4. Addition Property of Inequality
If the same number is added to both sides of a true inequality, the resulting inequality is also true. If 𝑎>𝑏, then 𝑎+𝑐>𝑏+𝑐 If 𝑎<𝑏, then 𝑎+𝑐<𝑏+𝑐

5. Example 1 pg. 285 Solve by adding
Solve 𝑐−12>65. Check your answers

6. Example 1 pg. 285 Solve by adding
Solve 𝑐−12>65. Check your answers 𝑐>77

7. Set-builder notation
A more concise way of writing a solution set. Example 1 answer would be written as 𝑥 𝑥>77

8. Subtraction Property of Inequality
If the same number is subtracted from each side of a true inequality, the resulting inequality is also true. If 𝑎>𝑏, then 𝑎−𝑐>𝑏−𝑐 If 𝑎<𝑏, then 𝑎−𝑐<𝑏−𝑐

9. Example 2 pg. 286 Solve by subtracting
Solve the inequality. 𝑥+23<14

10. Example 2 pg. 286 Solve by subtracting
Solve the inequality. 𝑥+23<14 𝑥 <−9

11. Example 3 pg. 287 Variables on each side.
Solve 12𝑛−4 < 13𝑛 Graph the solution set

12. Example 3 pg. 287 Variables on each side.
Solve 12𝑛−4 < 13𝑛 Graph the solution set 𝑛 𝑛 > −

13. Phrases for inequality
< Less than, fewer than > Greater than, more than < at most, no more than, less than or equal to > at least, no less than, greater than or equal to.

14. Example 4 pg. 287 Use an inequality to solve a problem
ENTERTAINMENT Tanya wants to buy season passes to two theme parks. If one season pass cost $54.99 and Tanya has $100 to spend on both passes, the second season pass must cost no more than what amount?

15. Example 4 pg. 287 Use an inequality to solve a problem
ENTERTAINMENT Tanya wants to buy season passes to two theme parks. If one season pass cost $54.99 and Tanya has $100 to spend on both passes, the second season pass must cost no more than what amount? The second season pass must cost no more than $45.01.

16. 5-1 pg o, 57-81(x3)

17. Solving Inequalities by Multiplication and Division
5-2 Notes for Algebra 1
Solving Inequalities by Multiplication and Division

18. Multiplication Property of Inequality
If both sides of an inequality that is true are multiplied by a positive number, the resulting inequality is also true. If 𝑎<𝑏 & 𝑐>0, then 𝑎𝑐<𝑏𝑐. If 𝑎>𝑏 & 𝑐>0, then 𝑎𝑐>𝑏𝑐. If both sides of an inequality that is true are multiplied by a negative number, the direction of the inequality is reversed to make the resulting inequality also true. If 𝑎<𝑏 & 𝑐<0, then 𝑎𝑐>𝑏𝑐. If 𝑎>𝑏 & 𝑐<0, then 𝑎𝑐<𝑏𝑐.

19. Example 1 pg. 293 write and solve an inequality
HIKING Mateo walks at a rate of mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Mateo to get there? Write and solve an inequality to find the time.

20. Example 1 pg. 293 write and solve an inequality
HIKING Mateo walks at a rate of 3 4 mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Mateo to get there? Write and solve an inequality to find the time. 3 4 𝑡 > 9; 𝑡 𝑡 > 12 ; It will take Mateo at least 12 hours.

21. Example 2 pg. 293 Solve by multiplying

22. Example 2 pg. 293 Solve by multiplying

23. Division Property of Inequalities
If both sides of an inequality that is true are divided by a positive number, the resulting inequality is also true. If 𝑎<𝑏 & 𝑐>0, then 𝑎 𝑐 < 𝑏 𝑐 . If 𝑎>𝑏 & 𝑐>0, then 𝑎 𝑐 > 𝑏 𝑐 . If both sides of an inequality that is true are divided by a negative number, the direction of the inequality is reversed to make the resulting inequality also true. If 𝑎<𝑏 & 𝑐<0, then 𝑎 𝑐 > 𝑏 𝑐 . If 𝑎>𝑏 & 𝑐<0, then 𝑎 𝑐 < 𝑏 𝑐 .

24. Example 3 pg. 294 Divide to solve the inequality
Solve each inequality 1.) 12𝑘 > 60 2.) −8𝑞<136

25. Example 3 pg. 294 Divide to solve the inequality
Solve each inequality 1.) 12𝑘 > 60 𝑘 𝑘 > 5 2.) −8𝑞<136 𝑞 𝑞≻17

26. 5-2 pg o, 48-66(x3)

27. Solving Multi-step inequalities
5-3 Notes for Algebra 1
Solving Multi-step inequalities

28. Example 1 pg. 298 Solve a Multi-Step inequality
FAXES Adriana has a budget of $115 for taxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 𝑝 < 115.

29. Example 1 pg. 298 Solve a Multi-Step inequality
FAXES Adriana has a budget of $115 for taxes. The fax service she uses charges $25 to activate an account and $0.08 per page to send faxes. How many pages can Adriana fax and stay within her budget? Use the inequality 𝑝 < 115. She can send at most 1125 pages

30. Example 2 pg. 298 Inequality involving a Negative Coefficient
Solve 13−11𝑑 > 79 Check your solution

31. Example 2 pg. 298 Inequality involving a Negative Coefficient
Solve 13−11𝑑 > 79 Check your solution 𝑑 𝑑 < −6

32. Example 3 pg. 299 Write and Solve an inequality
Define a variable, write an inequality, and solve the problem. Then check your solution. Four times a number plus twelve is less than the number minus three.

33. Example 3 pg. 299 Write and Solve an inequality
Define a variable, write an inequality, and solve the problem. Then check your solution. Four times a number plus twelve is less than the number minus three. Let 𝑛= the number; 4𝑛+12<𝑛−3; 𝑛 𝑛<−5

34. Example 4 pg. 299 Distributive Property
Solve 6𝑐+3 2−𝑐 > −2𝑐+1. Check your solution

35. Example 4 pg. 299 Distributive Property
Solve 6𝑐+3 2−𝑐 > −2𝑐+1. Check your solution 𝑐 𝑐 > −1

36. Example 5 pg. 300 Empty Set and all Reals
Solve each inequality. Check your solution. 1.) −7 𝑘+4 +11𝑘 > 8𝑘−2 2𝑘+1 2.) 2 4𝑟+3 < 22+8 𝑟−2

37. Example 5 pg. 300 Empty Set and all Reals
Solve each inequality. Check your solution. 1.) −7 𝑘+4 +11𝑘 > 8𝑘−2 2𝑘+1 ∅ 2.) 2 4𝑟+3 < 22+8 𝑟−2 𝑟 𝑟 is a real number

38. 5-3 pg o, 60-81(x3)

39. Solving Compound Inequalities
5-4 Notes for Algebra 1
Solving Compound Inequalities

40. Compound Inequality
Inequalities containing the words “and” or “or”

41. Intersection
A graph of “and” inequalities. Part of the graphs where the two solutions overlap

42. Example 1 pg. 306 Solve and Graph Intersection
Solve 7<𝑧+2 < 11. Then graph the solution set.

43. Example 1 pg. 306 Solve and Graph Intersection
Solve 7<𝑧+2 < 11. Then graph the solution set. 𝑧 5<𝑧 <

44. Union
Graph of “or” inequalities. “or” inequalities are true if at least one of the inequalities is true.

45. Example 2 pg. 307 Write and graph a compound inequality
TRAVEL A ski resort has several types of hotel rooms and cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount a guest would pay per night at the resort.

46. Example 2 pg. 307 Write and graph a compound inequality
TRAVEL A ski resort has several types of hotel rooms and cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount a guest would pay per night at the resort. 𝑛 𝑛 < 89 𝑜𝑟 𝑛 > 109 , where 𝑛 is the amount a guest pays per night

47. Example 3 pg. 308 Solve and Graph a Union
Solve 4𝑘−7 < 25 or 12−9𝑘 > 30. Then graph the solution set. 𝑘 𝑘 < 8 −

48. 5-4 pg o, 28-36, 45-72(x3)

49. Inequalities Involving Absolute Value
5-5 Notes for Algebra 1
Inequalities Involving Absolute Value

50. Cases for solving Absolute values inequalities <
Case 1: The expression inside the absolute value symbols is nonnegative Case 2: The expression inside the absolute value symbols is negative

51. Example 1 pg. 312 Solve Absolute Value Inequalities <
Solve each inequality and graph the solution set. 1.) 𝑛−3 < 12 2.) 𝑥+6 <−8

52. Example 1 pg. 312 Solve Absolute Value Inequalities <
Solve each inequality and graph the solution set. 1.) 𝑛−3 < 12 𝑛 −9 < 𝑛 <15 2.) 𝑥+6 <−8 ∅ −15 −9 3 −15 21

53. Example 2 pg. 313 Apply Absolute Value Inequalities
RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

54. Example 2 pg. 313 Apply Absolute Value Inequalities
RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California? 𝑥 13 < 𝑥 < 33

55. Cases for solving Absolute values inequalities >
Case 1: The expression inside the absolute value symbols is nonnegative Case 2: The expression inside the absolute value symbols is negative

56. Example 3 pg. 313 Solve Absolute Value Inequalities >
Solve each inequality then graph the solution set. 1.) 3𝑦−3 >9 2.) 2𝑥+7 > −11

57. Example 3 pg. 313 Solve Absolute Value Inequalities >
Solve each inequality then graph the solution set. 1.) 3𝑦−3 >9 𝑦 𝑦<−2 𝑜𝑟 𝑦>4 −5 − ) 2𝑥+7 > −11 𝑥 𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 −5 0 5

58. 5-5 pg o, 30, 31-35o, 48-66(x3)

59. Graphing Inequalities in Two variables
5-6 Notes for Algebra 1
Graphing Inequalities in Two variables

60. Boundary
An equation defines a BOUNDARY, which divides the plane into two half-planes. When the boundary is included it is a closed half-plane. When the boundary is not included, it is an open half-plane.

61. Graphing Linear Inequalities
1.) Graph the boundary. Use a Solid Line for < or > . Use a dashed line for < or >. 2.) Use a test point to determine which half-plane should be shaded. 3.) Shade the half-plane that contains the solution.

62. Example 1 pg. 317 Graph an Inequality <or>

63. Example 1 & 2 pg. 317 Graph an Inequality5 6 4 3 2 1 -4 -3 -2 -1

64. Example 3 pg. 318 Solve Inequalities from Graphs
Use a graph to solve 2𝑥+3 < 7.

65. Example 3 pg. 318 Solve Inequalities from Graphs
Use a graph to solve 2𝑥+3 < 7. 𝑥 < 2 5 4 3 2 1 -4 -3 -2 -1

66. Example 4 pg. 319 Write and Solve an Inequality
Joe writes and edits short articles for a local newspaper. It takes him about an hour to write and article and about a half-hour to edit an article In Joe works up to 8 hours a day , how many articles can he write and edit in one day?

67. Example 4 pg. 319 Write and Solve an Inequality
Joe writes and edits short articles for a local newspaper. It takes him about an hour to write and article and about a half-hour to edit an article In Joe works up to 8 hours a day , how many articles can he write and edit in one day? 𝑒 < −2𝑤+16

68. 5-6 pg , 44, 51-63(x3)