Disjunction : “Or” statement – Take the union of two solution sets! 2x + 5 < 3 or 1 – 2x...
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Expression: (no = sign)can be simplified or factored,
but NOT solved
Equation: two equal expressions (has = sign)
CAN be solved
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Equations:Conditional Equation: finite
solution setx2 – x – 6 = 0
Solution Set: { –2, 3}Identity: variable can be any real
number2(x – 3) = x – 6 + x
Solution Set: {reals}
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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, ...}
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Definition:Rational #: can be written as the
ratio of two integers where b ≠ 0.
That is: integers, fractions, terminating & repeating decimals.
Recall
ab
1 0.33
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Definition:Irrational #: a real number that is
not rational. (Duh!) That is: non-terminating, non-
repeating decimals like:0.12345678910111213141...
0.12122122212222..., ,, 2 5 e 2.718...
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Degree of an expression or equation:
The greatest power on any one term 5x7 + 11x5 – 7x3 + 2x (7th degree)
OR The greatest SUM of powers on any one term
5x2y3 + 11x2y7 – 7xy3 + 2 (9th degree)
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Disjunction: “Or” statement – Take the union of two solution
sets!
2x + 5 < 3 or 1 – 2x < 7
2x < – 2 – 2x < 6 x < – 1 OR x > 3
Solution Set: {x: x < – 1 or x > 3}
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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 < 4
Solution Set: {x: – 8 < x < 4}
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Absolute Value
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Absolute Value of a real number is the distance to the origin on the real number line.
Formal Definition:
00 0
0
x if xx if x
x if x
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The distance between two numbers a and b uses 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
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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|
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Check your understanding:
T F 1. |a| > 0T F 2. |a2| = a2
T F 3. |a3| = a3 T F 4. |a + b| = |a| + |b|T F 5. |ab| = |a| . |b|
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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)
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Absolute value equations may have zero, one or TWO
solutions:
Example 1:|a + 5| = 15
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Absolute value equations may have zero, one or TWO
solutions:Example 1:
|a + 5| = 15
a + 5 = 15 OR a + 5 = -15solution set: {10, -20}
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Absolute value equations may have zero, ONE or two
solutions:
Example 2: |a – 7.2| = 0
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Absolute value equations may have zero, ONE or two
solutions:
Example 2: |a – 7.2| = 0
a = 7.2 {7.2}
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Absolute value equations may have ZERO, one or two
solutions:
Example 3: |3a - 2| = -5
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Absolute value equations may have ZERO, one or two
solutions:
Example 3: |3a - 2| = -5
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Absolute value equations: Check your understanding.
Example 4:
4 33x
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Absolute value equations: Check your understanding.Example 4:
4 33x
{-21, -3}
4 3 4 33 3x xor
7 13 3x xor
21 3x or x
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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?
-3 30
-11 110
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Absolute Value Inequalities:1. Isolate the abs value sign on one
side of the equation.2. Separate into a disjunction or a
conjunction of two statements.3. Solve each statement alone.4. Combine to find the disjunction
or conjunction.
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Absolute Value Inequalities:
Example 1: | x + 2 | + 4 < 11
Isolate abs value first: | x + 2 | < 7
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Absolute Value Inequalities:
Example 1: | x + 2 | < 7
Begin by imagining: distance of some expression is less than 7 from origin!
-7 70
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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
-7 70
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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.
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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
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Absolute Value Inequalities:
Example 2: | 3x - 5 | > 2
Begin by picturing: distance is more than 2 units from the origin!
-2 20
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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
-2 20
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Absolute Value Inequalities:
Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7
3. Solve each statement alone.4. Combine to find the disjunction or conjunction.
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Absolute Value Inequalities:
Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7
{x: x<1 or x > }73
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Absolute Value Inequalities:
Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2
3x < 3 OR 3x >7
{x: x<1 or x > }*Hint: Abs value > pos # became
a DISJUNCTION!
73
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Absolute value inequalities: Check your understanding.
Example 3:
2 4 4 23
x
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Absolute value inequalities: Check your understanding.
Example 3: 2 4 4 23
x
4 33
4 3 4 33 3
7 13 3
21 3: 21 3
x
x xor
x xor
x or xx x or x
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Absolute value inequalities: Watch for special cases.
Example 4:
4 3x
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Absolute value inequalities: Watch for special cases.
Example 4: {real numbers}
*Absolute values are ALWAYS at least zero!
4 3x
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Absolute value inequalities: Watch for special cases.
Example 5: 4 3x
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Absolute value inequalities: Watch for special cases!
Example 5:
*Absolute values canNEVER be less than zero!
4 3x