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

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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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EquationsConditional Equation finite solution
setx2 x 6 0Solution Set 2,
3Identity variable can be any real number2(x
3) x 6 xSolution Set reals
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Number SetsCounting or Natural s1, 2, 3,
4, ...Whole s0, 1, 2, 3, 4,
...Integers...,-3, -2, -1, 0, 1, 2, 3, 4,
...
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DefinitionRational can be written as the
ratio of two integers where b ? 0. That is
integers, fractions, terminating repeating
decimals. Recall
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DefinitionIrrational a real number that is
not rational. (Duh!) That is non-terminating,
non-repeating decimals like0.1234567891011121314
1...0.12122122212222...
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Degree of an expression or equationThe
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 lt 3 or 1 2x lt 7
2x lt 2 2x lt 6 x lt 1 OR
x gt 3 Solution Set x x lt 1 or x gt 3
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Conjunction And statement Take the
intersection of two solution sets! 11lt 2x 5
lt1 3
11lt 2x 5 and 2x 5 lt1 3 16 lt 2x
2x lt 8 8 lt x AND x lt 4 Solution Set
x 8 lt x lt 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
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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 is4 -12 or -12 4 16 or -16
?16
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Formal Definition of Distance between two real
numbersThe 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 understandingT F 1. a gt 0T
F 2. a2 a2T F 3. a3 a3 T F
4. a b a bT F 5. ab a .
b
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Check your understandingF 1. a gt 0
(could be zero)T 2. a2 a2 (always
non-negative)F 3. a3 a3 (not when a lt
0)F 4. a b a b (e.g. when
a gt 0 and b lt 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 solutionsExample 1a 5 15
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Absolute value equations may have zero, one or
TWO solutionsExample 1 a 5 15a
5 15 OR a 5 -15solution set 10, -20
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Absolute value equations may have zero, ONE or
two solutionsExample 2 a 7.2 0
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Absolute value equations may have zero, ONE or
two solutionsExample 2 a 7.2 0a
7.2 7.2
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Absolute value equations may have ZERO, one or
two solutionsExample 3 3a - 2 -5
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Absolute value equations may have ZERO, one or
two solutionsExample 3 3a - 2 -5
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Absolute value equations Check your
understanding.Example 4
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Absolute value equations Check your
understanding.Example 4
-21, -3
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Absolute Value Inequalities

• Think Is the solution of x gt 11 a
  disjunction or a conjunction?
• Think Is the solution of x 3 a
  disjunction or a conjunction?

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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 lt 11
• Isolate abs value first x 2 lt 7

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Absolute Value Inequalities

• Example 1 x 2 lt 7
• Begin by imagining distance of some expression
  is less than 7 from origin!

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Absolute Value Inequalities

• Example 1 x 2 lt 7
• 2. Separate into a disjunction or a conjunction
  of two statements
• x 2 gt - 7 AND x 2 lt 7

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Absolute Value Inequalities

• Example 1
• x 2 lt 7
• x 2 gt - 7 AND x 2 lt 7
• x gt -9 AND x lt 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 lt 7
• x 2 gt - 7 AND x 2 lt 7
• x gt -9 AND x lt 5
• x -9 lt x lt 5
• Hint Abs Value lt Pos became a CONJUNCTION

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Absolute Value Inequalities

• Example 2 3x - 5 gt 2
• Begin by picturing distance is more than 2
  units from the origin!

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Absolute Value Inequalities

• Example 2 3x - 5 gt 2
• 2. Separate into a disjunction or a conjunction
  of two statements
• 3x - 5 lt - 2 OR 3x - 5 gt 2

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Absolute Value Inequalities

• Example 2 3x - 5 gt 2
• 3x - 5 lt - 2 OR 3x - 5 gt 2
• 3x lt 3 OR 3x gt7
• 3. Solve each statement alone.
• 4. Combine to find the disjunction or
  conjunction.

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Absolute Value Inequalities

• Example 2 3x - 5 gt 2
• 3x - 5 lt - 2 OR 3x - 5 gt 2
• 3x lt 3 OR 3x gt7
• x xlt1 or x gt

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Absolute Value Inequalities

• Example 2 3x - 5 gt 2
• 3x - 5 lt - 2 OR 3x - 5 gt 2
• 3x lt 3 OR 3x gt7
• x xlt1 or x gt
• Hint Abs value gt pos became a DISJUNCTION!

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Absolute value inequalities Check your
understanding.Example 3
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Absolute value inequalities Check your
understanding.Example 3
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Absolute value inequalities Watch for special
cases.Example 4
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Absolute value inequalities Watch for special
cases.Example 4 real
numbers Absolute values are ALWAYS at least
zero!
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Absolute value inequalities Watch for special
cases.Example 5
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Absolute value inequalities Watch for special
cases!Example 5 Absolute values
can NEVER be less than zero!