1. Inequalities

2. Inequalities and Their Graphs
Objective: To write, graph, and identify solutions of inequalities.

3. Vocabulary
An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two expressions.
You can use a number line to visually represent the values that satisfy an inequality
A solution of an inequality is any number that makes the inequality true.

4. Examples All real numbers x less than or equal to -7
6 less than a number k is greater than 13
All real numbers p greater than or equal to 1.5
The sum of t and 7 is less than -3

5. Examples

6. Solving Inequalities Using Addition or Subtractions
Objective: To use addition or subtraction to solve inequalities.

7. Vocabulary
Equivalent inequalities are inequalities that have the same solutions.
Just as you used the properties of equality to solve equations, you can use the properties of equality to solve inequalities.

8. Addition Property Let a, b, and c be real numbers
If a > b, then a + c > b + c
If a < b, then a + c < b + c
This property is also true for ≤ and ≥.
Example:
5 > 4, so > 4 + 3
3 < 7, so < 7 + 8

9. Subtraction Property Let a, b, and c be real numbers
If a > b, then a – c > b – c
If a < b, then a – c < b – c
This is also true for ≥ and ≤.
Example:
-3 < 5, so -3 – 2 < 5 – 2
3 > -4, so 3 – 3 > -4 – 3

10. Examples
X – 15 > -12
N – 5 < -3
M – 11 ≥ -2
10 ≤ x – 3

11. Examples
T + 6 > -4
-1 ≥ y + 12
85 + v ≤ 120
3.8 < b + 4

12. Examples
Your goal is to take at least 10,000 steps per day. According to your pedometer, you have walked 5274 steps. Write and solve an inequality to find the possible number of steps you can take to reach your goal.

13. Solving Inequalities Using Multiplication or Division
Objective: to use multiplication or division to solve inequalities.

14. Vocabulary
Just as you used multiplication and division to solve equations, you use multiplication and division to solve inequalities.
When dividing by a negative number, remember to change your sign! It is incorrect if you do not change your sign.

15. Multiplication Property
Let a, b, and c be real numbers with c > 0.
If a > b, then ac > bc.
If a < b, then ac < bc.
Example: -2 < 3, so -2(2) < 3(2)
Example: 4 > 1, s0 4(3) > 1(3)
Let a, b, and c be real numbers with c < 0.
If a > b, then ac < bc.
If a < b, then ac > bc.
Example: 3 > 1, so 3(-1) < 1(-1)
Example: 2 < 4, s0 2(-2) > 4(-2)

16. Division Property Let a, b, and c be real numbers with c > 0.
If a > b, then a/c > b/c.
If a < b, then a/c < b/c.
Example: 6 > 3, so 6/3 > 3/3
Example: 8 < 12, so 8/4 < 12/4
Let a, b, and c be real numbers with c < 0.
If a > b, then a/-c < b/-c.
If a < b, then a/-c > b/-c.
Example: 6 > 3, so 6/-3 < 3/-3
Example: 8 < 12, so 8/-4 > 12/-4

17. Examples
X / 5 ≥ -2
0 ≤ 3x / 11
4 > p / 8
W / 6 < 1

18. Examples
4t < -12
-30 > -5c
-4w ≤ 20
63 ≥ 7q

19. Examples
Text messages cost $0.15 each. You can spend no more than $10. How many text messages can you send?

20. Solving Multi-Step Inequalities
Objective: To solve multi-step inequalities.

21. Vocabulary
You solve a multi-step inequality in the same way you solve a one- step inequality.
You use the properties of inequality to transform the original inequality into a series of simpler, equivalent inequalities.

22. Examples 9 + 4t > 21 -6a – 7 ≤ 17 -4 < 5 – 3n 50 > 0.8x + 30
3(t + 1) – 4t ≥ -5
15 ≤ 5 – 2(4m + 7)
6n – 1 > 3n + 8
3b + 12 > 27 – 2b

23. Compound Inequalities
Objective: To solve and graph inequalities containing the word and. To solve and graph inequalities containing the word or.

24. Vocabulary
A compound inequality consists of two distinct inequalities joined by the word and or the word or.
You find the solutions of a compound inequality either by identifying where the solution sets of the distinct inequalities overlap or by combining the solution sets to form a larger solution set.

25. Vocabulary
The graph of a compound inequality with the word and contains the overlap of the graphs of the two inequalities that form the compound inequality.
Compound inequalities that use the word and can also use the word inclusive meaning that the inequality includes both inequalities.
A solution of a compound inequality involving and is any number that makes both inequalities true.

26. Writing a Compound Inequality
All real numbers that are greater than -2 and less than 6
All real numbers that are greater than or equal to -4 and less than 8
The circumference of a women’s basketball much be between inches and 29 inches, inclusive

27. Examples -3 ≤ m – 4 < 1 -2 < 3y – 4 < 14 5 ≤ y + 2 ≤ 11

28. Examples 2 ≤ 0.75w ≤ 4.5 -3 ≤ (6 – q) / 9 ≤ 3 -4 < k + 3 < 8

29. Examples
To earn a B in your algebra course, you must achieve an unrounded test average between 84 and 86, inclusive. You scored 86, 85, and 80 on the first three tests of the grading period. What possible scores can you earn on the fourth and final test to earn a B in the course?

30. Examples
Suppose you scored 78, 78, and 79 on the first three tests. Is it possible for you to earn a B in the course? Assume that 100 is the maximum grade you can earn in the course and on the test.

31. Vocabulary
The graph of a compound inequality with the word or contains each graph of the two inequalities that form the compound inequality.
A solution of a compound inequality involving or is any number that makes either inequality true.

32. Writing a Compound Inequality
All real numbers that are less than 0 or greater than or equal to 5
All real numbers that are less than or equal to 2.5 or greater than 6
All real numbers that are less than -5 and greater than 7

33. Examples 3t + 2 < -7 or -4t + 5 < 1
-2 y + 7 <1 or 4y + 3 ≤ -5
6b – 1 < -7 or 2b + 1 > 5
4d + 5 ≥ 13 or 7d – 2 < 12

34. Examples 5y + 7 ≤ -3 or 3y – 2 ≥ 13 5 + m > 4 or 7m < -35
7 – c < 1 or 4c ≤ 12
5z – 3 > 7 or 4z – 6 < -10

35. Vocabulary
You can use an inequality to describe a portion of the number line called an interval.
You can also use interval notation to describe an interval on the number line.
Interval notation include the use of three special symbols.

36. Interval Notation Symbols
Parentheses: Use ( or ) when a < or > symbol indicates that the interval’s endpoints are NOT included.
Brackets: Use [ or ] when a ≤ or ≥ symbol indicates that the interval’s endpoints are included.
Infinity: Use ∞ when the interval continues forever in a positive direction. Use -∞ when the interval continues forever in a negative direction.
Always use parentheses around (-∞ and ∞)

37. Examples
X > 2
X ≤ 2
1 < x ≤ 5
X < -3 or x ≥ 4

38. Examples -4 ≤ x < 6 X ≤ 1 or x > 2 -9 < 3m + 6 ≤ 18
F + 14 < 9 or -9f ≤ -45

39. Examples
(-2, 7]
(4, ∞)
(-∞, 2]
[-4, 5]

40. Examples (-∞, -1] or (3, ∞) [6, ∞) (-∞, 4) or [ 4.5, ∞)

41. Absolute Value Equations and Inequalities
Objective: To solve equations and inequalities involving absolute value.

42. Vocabulary
You can solve absolute value equations and inequalities by first isolating the absolute value expression, if necessary. Then write an equivalent pair of linear equations or inequalities.
Recall that absolute value represents distance from 0 on a number line. Distance is always nonnegative. So any equation that states the absolute value of an expression is negative has no solutions.
To solve an equation in the form ΙAΙ = b, where A represents a variable expression and b > 0, solve A = b and A = -b.

43. Review
Ι-9Ι
Ι300Ι
Ι4 - 9Ι
Ι-7 – 3Ι

44. Examples
ΙxΙ= 12
ΙbΙ= 3
ΙyΙ = -4
ΙrΙ= - 25

45. Examples ΙxΙ+ 2 = 9 ΙnΙ– 5 = -2 Ι2x -5Ι= 13 Ι100 – 2tΙ= 40
3Ι2z + 9Ι+ 12 = 10
Ι3x – 6Ι– 5 = -7
4Ι2y – 3Ι– 1 = 11

46. Vocabulary
To solve an inequality in the form ΙAΙ< b, where A is a variable expression and b > 0, solve the compound inequality –b < A < b. (THIS IS AN AND INEQUALITY!)
To solve an inequality in the form ΙAΙ> b, where A is a variable expression and b > 0, solve the compound inequality A < -b or A > b. (THIS IS AN OR INEQUALITY!)
Similar rules are true for ΙAΙ ≤ b and ΙAΙ ≥ b.

47. Examples ΙxΙ< 5 Ιy – 2Ι≤ 1 Ι2f + 9Ι≤ 13 3Ι2v – 1Ι< 27
ΙnΙ– 3 < -1
-7Ι3vΙ< 28

48. Examples ΙxΙ≥ 3 Ιy + 8Ι> 3 Ι4w + 1Ι> 11 4Ι5t – 4Ι≥ 16
2Ι9c – 13Ι> -20

49. Examples
A company makes boxes of crackers that should weigh 213 grams. A quality-control inspector randomly selects boxes to weigh. Any box that varies from the weight by more than 5 grams is sent back. What is the range of allowable weights for a box of crackers?

50. Examples
A food manufacturer makes 32 ounce boxes of pasta. Not every box weighs exactly 32 ounces. The allowable difference from the ideal weight is at most 0.05 ounces. Write and solve an absolute value inequality to find the range of allowable weights.