1. Inequalities and Proof
Chapter 2
Inequalities and Proof

2. Solving Inequalities in One Variable
Section 2-1
Solving Inequalities in One Variable

3. Transitive Property - If a < b and b < c, then a < c
Properties of Order
Transitive Property - If a < b and b < c, then a < c
Addition Property - If a < b, then a + c < b + c

4. 1. If a < b and c is positive, then ac < bc
Multiplication Property
1. If a < b and c is positive, then ac < bc
2. If a < b and c is negative, then ac > bc

5. Equivalent Inequalities
Inequalities with the same solution set
2x + 5 < 13 and 2x < 8 and x < 4
4x > 2(3 + 2x) and 2x > 3 + 2x

6. Transformations that Produce Equivalent Inequalities
Simplifying either side of an inequality.

7. Transformations that Produce Equivalent Inequalities
Adding to (or subtracting from) each side of an inequality, the same number or the same expression.

8. Transformations that Produce Equivalent Inequalities
Multiplying (or dividing) each side of an inequality by the same negative number and reversing the inequality.

9. Transformations that Produce Equivalent Inequalities
Multiplying (or dividing) each side of an inequality by the same positive number

10. Examples
Solve each inequality and graph its solution set
5x + 17 < 2
5(3-t) < 7 - t

11. Solving Combined Inequalities
Section 2-2
Solving Combined Inequalities

12. Conjunction- Example:
A sentence formed by joining two sentences with the word and. In a conjunction both sentences are true.
Example:
Graph the solution set of the conjunction x > -2 and x < 3

13. Disjunction-
A sentence formed by joining two sentences with the word or. It is true when at least one of the sentences is true.
Example:
Graph the solution set for the disjunction x < 2 or x = 2

14. Conjunctions in a Different form
Solve 3 < 2x + 5 ≤ 15.
First rewrite the conjunction with and.
3 < 2x + 5 and 2x + 5 ≤ 15
Now solve each inequality and graph the solution set for the conjunction.

15. Conjunctions in a Different form
Solve -3 < -2(t -3) < 6.
First rewrite the conjunction with and.
-3 < -2(t-3) and -2(t-3) < 6
Now solve each inequality and graph the solution set for the conjunction.

16. 2t + 7  13 or 5t – 4 < 6 Disjunctions
Solve 2t + 7  13 or 5t – 4 < 6.
Now solve each inequality and graph the solution set for the disjunction.
2t + 7  13 or 5t – 4 < 6

17. y  -1 or y  3 Solve y  -1 or y  3 Disjunctions
Now solve each inequality and graph the solution set for the disjunction.
y  -1 or y  3

18. Problem Solving Using Inequalities
Section 2-3
Problem Solving Using Inequalities

19. Solving Word Problems Using Inequalities
Phrase Translation
x is at least a
x is no less than a
x ≥ a
x is at most b
x is no greater than b
x ≤ b
x is between a and b
x is between a and b, inclusive
a < x < b
a ≤ x ≤ b

20. Example:
Find all sets of 4 consecutive integers whose sum is between 10 and 20.

21. Solution Four consecutive integers – n + (n + 1) + (n + 2) + (n + 3)
Which integers work?

22. Absolute Value in Open Sentences
Section 2-4
Absolute Value in Open Sentences

23. Absolute Value
The distance between a number x and zero on a number line
If | x | = 1, then x = 1 or -1
If | x | < 1, then -1 < x < 1
If | x | > 1, then x < -1 or x > 1

24. Example - Equality To solve, set up two equations Solve |3x - 2| = 8
3x – 2 = – x – 2 = 8
3x = – x = 10
x = – x = 10/3
The solution is {-2, 10/3}

25. The solution set is { t: – 1 < t < 4}
Example - Inequality
Solve |3 – 2t| < 5
Set up a compound inequality
– 5 < 3 – 2t < 5
– 8 < – 2t < 2
4 > t > – 1
The solution set is { t: – 1 < t < 4}

26. Solving Absolute Value Sentences Graphically
Section 2-5
Solving Absolute Value Sentences Graphically

27. Facts The distance between x and 0 on a number line is | x |
The distance between the graphs of real numbers a and b is | a – b |, or | b – a |

28. Examples
Solve |5 - x| = 2
{3, 7}

29. Examples
Solve |b + 5| > 3
{b: b < -8 or b > -2}

30. Examples
Solve |2n + 5| ≤ 3
{n: n ≤ -4 or n ≥ -1}

31. Section 2-6
Theorems and Proofs

32. Definitions Theorem - A statement that can be proved
Corollary – A theorem that can be proved easily from another
Axioms – Statements that we assume to be true (these are also called postulates)

33. Cancellation Property of Addition
For all real numbers a, b, and c:
If a + c = b + c, then a = b
If c + a = c + b, then a = b

34. Cancellation Property of Multiplication
For all real numbers a and b, and nonzero real numbers c:
If ac = bc, then a = b
If ca = cb, then a = b

35. Zero – Product Property
For all real numbers a and b:
ab = 0 if and only if a = 0 or b = 0

36. Theorems about Order and Absolute Value
Section 2-7
Theorems about Order and Absolute Value