Disjunction : “Or” statement – Take the union of two solution sets! 2x + 5 < 3 or 1 – 2x < 7

1. Expression: (no = sign)can be simplified or factored, but NOT solvedEquation: two equal expressions (has = sign) CAN be solved

2. Equations:Conditional Equation: finite solution setx2 – x – 6 = 0Solution Set: { –2, 3}Identity: variable can be any real number2(x – 3) = x – 6 + xSolution Set: {reals}

3. Number Sets:Counting or Natural #s:{1, 2, 3, 4, ...}Whole #s:{0, 1, 2, 3, 4, ...}Integers:{...,-3, -2, -1, 0, 1, 2, 3, 4, ...}

4. Definition:Rational #: can be written as the ratio of two integers where b ≠ 0.That is: integers, fractions, terminating & repeating decimals. Recall

5. Definition:Irrational #: a real number that is not rational. (Duh!)That is: non-terminating, non-repeating decimals like:0.12345678910111213141...0.12122122212222...

6. Degree of an expression or equation:The greatest power on any one term5x7 + 11x5 – 7x3 + 2x (7th degree)OR The greatest SUM of powers on any one term5x2y3 + 11x2y7 – 7xy3 + 2 (9th degree)

7. Disjunction: “Or” statement – Take the union of two solution sets!2x + 5 < 3 or 1 – 2x < 7 2x < – 2 – 2x < 6x < – 1 OR x > 3Solution Set: {x: x < – 1 or x > 3}

8. Conjunction: “And” statement – Take the intersection of two solution sets! – 11< 2x + 5 <1 3 – 11< 2x + 5 and 2x + 5 <1 3 – 16 < 2x 2x < 8–8 < x AND x < 4Solution Set: {x: – 8 < x < 4}

9. Absolute Value

10. Absolute Value of a real number is the distance to the origin on the real number line.Formal Definition:

11. The distance between two numbers a and buses absolute value because we can subtract in either order and then make the answer positive(distances are never negative).e.g. Distance between 4 and -12 is|4 – -12| or |-12 – 4| |16| or |-16| 16

12. Formal Definition of Distance between two real numbers:The distance between a and b is given by the absolute value of the difference of the coordinates.Distance between a and b = |a – b| or |b – a|

13. Check your understanding:T F 1. |a| > 0T F 2. |a2| = a2T F 3. |a3| = a3T F 4. |a + b| = |a| + |b|T F 5. |ab| = |a| . |b|

14. Check your understanding:F 1. |a| > 0 (could be zero)T 2. |a2| = a2(always non-negative)F 3. |a3| = a3(not when a < 0)F 4. |a + b| = |a| + |b|(e.g. when a > 0 and b < 0)T 5. |ab| = |a| . |b|(makes it positive sooner or later)

15. Absolute value equations may have zero, one or TWO solutions:Example 1:|a + 5| = 15

16. Absolute value equations may have zero, one or TWO solutions:Example 1: |a + 5| = 15a + 5 = 15 OR a + 5 = -15solution set: {10, -20}

17. Absolute value equations may have zero, ONE or two solutions:Example 2: |a – 7.2| = 0

18. Absolute value equations may have zero, ONE or two solutions:Example 2: |a – 7.2| = 0a = 7.2 {7.2}

19. Absolute value equations may have ZERO, one or two solutions:Example 3: |3a - 2| = -5

20. Absolute value equations may have ZERO, one or two solutions:Example 3: |3a - 2| = -5

21. Absolute value equations: Check your understanding.Example 4:

22. Absolute value equations: Check your understanding.Example 4: {-21, -3}

23. -11 0 11 -3 0 3 Absolute Value Inequalities: Think: Is the solution of | x | > 11 a disjunction or a conjunction? Think: Is the solution of | x | ≤ 3 a disjunction or a conjunction?

24. Absolute Value Inequalities: • Isolate the abs value sign on one side of the equation. • Separate into a disjunction or a conjunction of two statements. • Solve each statement alone. • Combine to find the disjunction or conjunction.

25. Absolute Value Inequalities: Example 1: | x + 2 | + 4 < 11 Isolate abs value first: | x + 2 | < 7

26. 0 -7 7 Absolute Value Inequalities: Example 1: | x + 2 | < 7 Begin by imagining: distance of some expression is less than 7 from origin!

27. 0 -7 7 Absolute Value Inequalities: Example 1: | x + 2 | < 7 2. Separate into a disjunction or a conjunction of two statements x + 2 > - 7 AND x + 2 < 7

28. Absolute Value Inequalities: Example 1: | x + 2 | < 7 x + 2 > - 7 AND x + 2 < 7 x > -9 AND x < 5 3. Solve each statement alone. 4. Combine to find the disjunction or conjunction.

29. Absolute Value Inequalities: Example 1: | x + 2 | < 7 x + 2 > - 7 AND x + 2 < 7 x > -9 AND x < 5 {x: -9 < x < 5} *Hint: Abs Value < Pos # became a CONJUNCTION

30. 0 -2 2 Absolute Value Inequalities: Example 2: | 3x - 5 | > 2 Begin by picturing: distance is more than 2 units from the origin!

31. 0 -2 2 Absolute Value Inequalities: Example 2: | 3x - 5 | > 2 2. Separate into a disjunction or a conjunction of two statements 3x - 5 < - 2 OR 3x - 5 > 2

32. Absolute Value Inequalities: Example 2: | 3x - 5 | > 2 3x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7 3. Solve each statement alone. 4. Combine to find the disjunction or conjunction.

33. Absolute Value Inequalities: Example 2: | 3x - 5 | > 2 3x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7 {x: x<1 or x > }

34. Absolute Value Inequalities: Example 2: | 3x - 5 | > 2 3x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7 {x: x<1 or x > } *Hint: Abs value > pos # became a DISJUNCTION!

35. Absolute value inequalities: Check your understanding.Example 3:

36. Absolute value inequalities: Check your understanding.Example 3:

37. Absolute value inequalities: Watch for special cases.Example 4:

38. Absolute value inequalities: Watch for special cases.Example 4: {real numbers}*Absolute values are ALWAYS at least zero!

39. Absolute value inequalities: Watch for special cases.Example 5:

40. Absolute value inequalities: Watch for special cases!Example 5: *Absolute values can NEVER be less than zero!