Thinking Mathematically

1
Thinking Mathematically

• Algebra 1
• By A.J. Mueller

2
Properties
3
Proprieties

• Addition Property (of Equality)
• 459
• Multiplication Property (of Equality)
• 5?840
• Reflexive Property (of Equality)
• 1212
• Symmetric Property (of Equality)
• If ab then ba

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Proprieties

• Transitive Property (of Equality)
• If ab and bc then ac
• Associative Property of Addition
• (0.65.3)4.70.6(5.34.7)
• Associative Property of Multiplication
• (-5?7) 3-5(7?3)
• Commutative Property of Addition
• 2xx2

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Proprieties

• Communicative Property of Multiplication
• b3a2a2b3
• Distributive Property
• 5(2x7) 10x35
• Prop. of Opposites or Inverse Property of
  Addition
• a(-a)0 and (-a)a0
• Prop. of Reciprocals or Inverse Prop. of
  Multiplication
• x2/77/x21

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Proprieties

• Identity Property of Addition
• -50-5
• Identity Property of Multiplication
• x?1x
• Multiplicative Property of Zero
• 5?00
• Closure Property of Addition
• For real a and b, ab is a real number

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Proprieties

• Closure Property of Multiplication
• ab ba
• Product of Powers Property
• x3x4x7
• Power of a Product Property
• (pq)7p7q7
• Power of a Power Property
• (n2) 3

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Proprieties

• Quotient of Powers Property
• X5/x3x2
• Power of a Quotient Property
• (a/b) 2
• Zero Power Property
• (9ab)01
• Negative Power Property
• h-21/h2

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Proprieties

• Zero Product Property
• ab0, then a0 or b0
• Product of Roots Property
• v20 v4v5
• Power of a Root Property
• (v7) 27

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Solving 1st Power Inequalities in One Variable
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Solving 1st Power Inequalities in One Variable

• With only one inequality sign

x gt -5
Solution Set x x gt -5
Graph of the Solution
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Conjunctions

• Open endpoint for these symbols gt lt
• Closed endpoint for these symbols or
• Conjunction must satisfy both conditions
• Conjunction AND

x -4 lt x 9
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Disjunctions

• Open endpoint for these symbols gt lt
• Closed endpoint for these symbols or
• Disjunction must satisfy either one or both of
  the conditions
• Disjunction OR

x x lt -4 or x 7
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Special Cases That All Reals

• Watch for special cases
• No solutions that work Answer is Ø
• Every number works Answer is reals
• Disjunction in same direction answer is one arrow

x x gt -5 or x 1
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Special Cases That
x -x lt -2 and -5x 15
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Linear equations in two variables
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Linear equations in two variables

• Lots to cover here slopes of all types of
  lines equations of all types of lines,
  standard/general form, point-slope form, how to
  graph, how to find intercepts, how and when to
  use the point-slope formula, etc. Remember you
  can make lovely graphs in Geometer's Sketchpad
  and copy and paste them into PPT.

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Important Formulas

• Slope-
• Standard/General form- axbxc
• Point-slope form-
• Use point-slope formula when you know 4 points on
  2 lines.
• Vertex-
• X-intercepts- set f(x) to 0 then solve
• Y-intercepts- set the x in the f(x) to 0 and then
  solve

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Examples of Linear Equations

• Example 1
• y-3/4x-1

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Examples of Linear Equations

• Example 2
• 3x-2y6 (Put into standard form)
• 2y-3x6 (Divide by 2)
• y-3/2x6 (Then graph)

21
Linear Systems
22
Substitution Method

• Goal replace one variablewith an equal
  expression

Step 1 Look for a variable with a coefficient
of one. Step 2 Isolate that variable Equation
A now becomes y 3x 1 Step 3 SUBSTITUTE
this expression into that variable in Equation
B Equation B now becomes 7x 2( 3x 1 ) -
4 Step 4 Solve for the remaining variable Step
5 Back-substitute this coordinate into Step 2 to
find the other coordinate. (Or plug into any
equation but step 2 is easiest!)
23
Addition/ Subtraction (Elimination) Method

• Goal Combine equations to cancel out one
  variable.

Step 1 Look for the LCM of the coefficients on
either x or y. (Opposite signs are recommended
to avoid errors.) Here -3y and 2y could be
turned into -6y and 6y Step 2 Multiply each
equation by the necessary factor. Equation A now
becomes 10x 6y 10 Equation B
now becomes 9x 6y -48 Step 3 ADD the two
equations if using opposite signs (if not,
subtract) Step 4 Solve for the remaining
variable Step 5 Back-substitute this coordinate
into any equation to find the other coordinate.
(Look for easiest coefficients to work with.)
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Factoring
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Types of Factoring

• Greatest Common Factor (GFC)
• Difference of Squares
• Sun and Difference of Cubes
• Reverse FOIL
• Perfect Square Trinomial
• Factoring by Grouping (3x1 and 2x2)

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GFC

• To find the GCF, you just look for the variable
  or number each of the numbers have in common.
• Example 1
• x25x15
• x(2515)

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Difference of Squares

• Example 1
• 27x475y4
• 3(9x425y4)
• 3(3x25y2)(3x2-5y2)
• Example 2
• 45x6-81y4
• 9(5x4-9y4)

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Sun and Difference of Cubes

• Example 1
• (8x327)
• (2x3)
• (4x2-6x9)
• Example 2
• (p3-q3)
• (p-q) (p2pqq2)

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Reverse FOIL

• Example 1
• x2-19x-32
• (x8)(x-4)
• Example 2
• 6y2-15y12
• (3y-4)(2y-3)

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Perfect Square Trinomial

• Example 1
• 4y230y25
• (2y5) 2
• Example 2
• x2-10x25
• (x-5) 2

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Factoring By Grouping

• 3x1
• Example 1
• a24a4-b2
• (a4a4)-(b2)
• (a2)-(b2)
• (a2-b)(a2b)

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Factoring by Grouping

• 2x2
• Example 1
• 2xy24x4y
• xy2y4xy

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Quadratic Equations
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Factoring Method

• Set equal to zero
• Factor
• Use the Zero Product Property to solve.
• Each variable equal to zero.

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Factoring Method Examples

• Any of terms- look for GCF first
• Example 1
• 2x28x (subtract 8x to set equation equal to
  zero)
• 2x2-8x0 (now factor out the GCF)
• 2x(x-4)0

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Factoring Method Examples

• Set 2x0, divide 2 on both sides and x0
• Set x-40, add 4 to both sides and x4
• x is equal to 0 or 4
• The answer is 0,4

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Factoring Method- Binomials

• Binomials Look for Difference of Squares
• Example 1
• x281 (subtract 81 from both sides)
• x2-810 (factoring equation into conjugates)
• (x9)(x-9)0
• x90 or x-90

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Factoring Method- Binomials

• x90 (subtract 9 from both sides)
• x-9
• x-90 (add 9 to both sides)
• x9
• The answer is -9,9

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Factoring Method-Trinomials

• Trinomials Look for PST
• Example 1
• x2-9x-18 (add 18 to both sides)
• x2-9x180 (x2-9x18 is a PST)
• (x-9)(x-9)0
• x-90 (add 9 to both sides) x9
• The answer is 9d.r. d.r.- double root

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Square Roots of Both Sides

• Reorder terms IF needed
• Works whenever form is (glob)2 c
• Take square roots of both sides
• Simplify the square root if needed
• Solve for x, or in other words isolate x.

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Square Roots Of Both Sides

• Example 1
• (Factor out the GCF)
• 2(x2-6x-2)0 (You can get rid of the 2 because it
  does not play a role in this type of equation)
• x2-6-2x0 (Add the 2 to both sides)
• x2-6x__2__ (Take half of the middle number which
  right now is 6)
• x2-6x929 (Simplify)

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Square Roots Of Both Sides

• (x-3)11 (Then take the square root of both
  sides)
• (x-3) 11 (Continue to simplifying)
• (Add the 3 to both sides)
• (Final Answer)

43
Completing the Square

• Example 1
• 2x2-12x-40 (Factor out the GCF)
• 2(x2-6x-2)0 (You can get rid of the 2 because it
  does not play a role in this type of equation)
• x2-6-2x0 (Add the 2 to both sides)
• x2-6x__2__ (Take half of the middle number which
  right now is 6)
• x2-6x929 (Simplify)

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Completing the Square

• (x-3)11 (Then take the square root of both
  sides)
• v(x-3) /-v11 (Continue to simplifying)
• (x-3)/- v11 (Add the 3 to both sides)
• x3/- v11 (Final Answer)

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Quadratic Formula

• This is a formula you will need to memorize!
• Works to solve all quadratic equations
• Rewrite in standard form in order to identify the
  values of a, b and c.
• Plug a, b c into the formula and simplify!
• QUADRATIC FORMULA

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Quadratic Formula Examples

• Example 1
• 3x2-6x212x
• Put this in standard form 2x2-12x-60
• Put into quadratic formula

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Quadratic Formula Examples
48
The Discriminant Making Predictions

• b2-4ac2 is called the discriminant
• Four Cases

1. b2 4ac positive non-square? two irrational
roots
2. b2 4ac positive square? two rational roots
3. b2 4ac zero? one rational double root
4. b2 4ac negative? no real roots
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The Discriminant Making Predictions
Use the discriminant to predict how many roots
each equation will have.
1. x2 7x 2 0
494(1)(-2)57 ?2 irrational roots
2. 0 2x2 3x 1
94(2)(1)1 ? 2 rational roots
3. 0 5x2 2x 3
44(5)(3)-56 ? no real roots
4. x2 10x 250
1004(1)(25)0 ? 1 rational double root
50
The Discriminant Making Predictions

• The zeros of a function are the x-intercepts on
  its graph. Use the discriminant to predict how
  many x-intercepts each parabola will have and
  where the vertex is located.

1. y 2x2 x - 6
14(2)(-6)49 ? 2 rational zeros opens up/vertex
below x-axis/2 x-intercepts
2. f(x) 2x2 x 6
14(2)(6)-47 ? no real zeros opens up/vertex
above x-axis/No x-intercepts
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The Discriminant Making Predictions

814(-2)(6)129 ?2 irrational zeros opens
down/vertex above x-axis/2 x-intercepts
3. y -2x2 9x 6
4. f(x) x2 6x 9
364(1)(9)0 ? one rational zero opens up/vertex
ON the x-axis/1 x-intercept
I (A.J. Mueller) got these last four slides from
Ms. Hardtkes Power Point of the Quadratic
Methods.
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Functions
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About Functions

• Think of f(x) like y, they are really the same
  thing.
• The domain is the x line of the graph
• The Range is the y line of the graph

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Functions

• f(x) -2x-8
• First find the vertex.
• ( ) The vertex of this equation is (1,-9)
• Find the x-intercepts by setting f(x) to 0. The
  x-intercepts are -2,4
• Find the y-intercept by setting the x in the f(x)
  to 0. You would get -8.
• The graph the equation.

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Simplifying expressions with exponents

• This site will example how to simplify
  expressions with exponents very well.
• http//www.purplemath.com/modules/simpexpo.htm

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Radicals

• Example 1
• (Simplify)
• (Now you can cancel the v2s)

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Radicals

• Example 2
• (Multiply by )
• That equals
• Cancel out the 2s and the final answer is

58
Radicals

• Example 3
• Take the square root of that.
• Final answer is

59
Word Problems

• Example 1
• If Tom weighs 180 on the 3th day of his diet and
  166 on the 21st day of his diet, write an
  equation you could use to predict his weight on
  any future day.
• (day, weight)
• (3,180)
• 21,166)

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Word Problems

• Point Slope m166-180/21-31
• That can be simplified to -14/18 and then -7/9.
• 4-180-7/9(x-3)
• 4-180-7/921/9
• Answer y-7/9x182 1/3

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Word Problems

• Click to open the hyperlink. Then try out this
  quadratic word problem, it will walk you through
  the process of finding the answer.
• http//www.algebra.com/algebra/homework/quadratic/
  word/02-quadratic.wpm

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Word Problems

• Here is another link to a word problem about time
  and travel.
• http//www.algebra.com/algebra/homework/word/trave
  l/07-cockroach.wpm

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Word Problems

• This word problem is about geometry.
• http//www.algebra.com/algebra/homework/word/geome
  try/02-rectangle.wpm
• This site is good study tool for word problems.

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Line of Best Fit

• The Line of Best Fit is your guess where the
  middle of all the points are.
• http//illuminations.nctm.org/ActivityDetail.aspx?
  id146
• This URL is a good site to example Line of Best
  Fit. Plot your points, guess your line of best
  fit, then the computer will give the real line of
  best fit.

65
Line of Best Fit

• Your can use a Texas Instruments TI-84 to graph
  your line of best fit and also all other types of
  graphs.