Algebra 1-semester exam review

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Algebra 1-semester exam review

• By Ricardo Blanco

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In the next slides you will review

• Addition Property (of Equality)
• Multiplication Property (of Equality)
• Reflexive Property (of Equality)
• Symmetric Property (of Equality)
• Transitive Property (of Equality)

• The properties we learned
• What are they used for and when to recognize them

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Properties

• Examples in order
• 1. if a b, then a c b c.
• is added to both sides of an equation, the two
  sides remain equal. That is,
• 2.if a b, then a c b c.
• . If the same number If  a b  then  ac bc.

• 1.Addition Property (of Equality)
• 2. Multiplication Property (of Equality)

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Properties

• 3. Reflexive Property (of Equality)
• 4. Symmetric Property (of Equality)
• 5. Transitive Property (of Equality)

• 3. aa
• 4. if ab then ba
• 5. If a b and b c, then a c.

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In the next slides you will review

• Associative Property of Addition
• Associative Property of Multiplication
• Commutative Property of Addition
• Commutative Property of Multiplication
• Distributive Property (of Multiplication over
  Addition)

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Properties

• 6. Associative Property of Addition
• 7. Associative Property of Multiplication

• 6. the sum does not change. (2 5) 4 11 or 2
  (5 4) 11
• 7. answer will still not chage.(3 x 2) x 4 24
  or 3 x (2 x 4) 24.

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Properties

• 8. Commutative Property of Addition
• 9. Commutative Property of Multiplication

• 8. As per the commutative property of addition,
  the expression 5 14 19 can be written as 14
  5 19. so, 5 14 14 5.
• 9. 4 x 2 2 x 4

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Properties

• 10. Distributive Property (of Multiplication over
  Addition)

• 10. 3(2 7 - 5)     3(2) 3(7) (3)(-5)
• 3(4)   6      21 - 15
•                                                  
  12     12

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In the next slides you will review

• Prop of Opposites or Inverse Property of Addition
• Prop of Reciprocals or Inverse Prop. of
  Multiplication
• Identity Property of Addition
• Identity Property of Multiplication

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Properties

• 11. Prop of Opposites or Inverse Property of
  Addition
• 12. Prop of Reciprocals or Inverse Prop. of
  Multiplication

• 11. In other words, when you add a number to its
  additive inverse, the result is 0. Other terms
  that are synonymous with additive inverse are
  negative and opposite. a (-a) 0.
• 12. In other words, when you multiply a number by
  its multiplicative inverse the result is 1.  A
  more common term used to indicate a 
  multiplicative inverse is the reciprocal.  A
  multiplicative inverse or reciprocal of a real
  number a (except 0) is found by "flipping" a
  upside down.  The numerator of a becomes the
  denominator of the reciprocal of a and the
  denominator of a becomes the numerator of the
  reciprocal of a.

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Properties

• 13. Identity Property of Addition
• 14. Identity Property of Multiplication

• 13. Identity property of addition states that the
  sum of zero and any number or variable is the
  number or variable itself. 4 0 4
• 14. According to identity property of addition,
  the sum of a number and 0 is the number itself.  
  4 1 4

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In the next slides you will review

• Multiplicative Property of Zero
• Closure Property of Addition
• Closure Property of Multiplication
• Product of Powers Property
• Power of a Product Property
• Power of a Power Property

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Properties

• 15. Multiplicative Property of Zero
• 16. Closure Property of Addition
• 17. Closure Property of Multiplication

• 15. The product of any number and zero is zero- a
  0 0
• 16. Closure property of addition states that the
  sum of any two real numbers equals another real
  number.
• 17. Closure property of multiplication states
  that the product of any two real numbers equals
  another real number.

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Properties

• 18. Product of Powers Property
• 19. Power of a Product Property
• 20. Power of a Power Property

• 18.when you multiply powers having the same
  amount add the exponents.
• 72 76
• (7 7) (7 7 7 7 7 7)
• 19. (3t)4
• (3t)4 34 t4 81t4
• 20. (ab)c abc

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In the next slides you will review

• Quotient of Powers Property
• Power of a Quotient Property
• Zero Power Property
• Negative Power Property
• zero product property

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Properties

• 21. Quotient of Powers Property
• 22. Power of a Quotient Property

• 21. This property states that to divide powers
  having the same base, subtract the
  exponents.(am)n amn
• 22. This property states that the power of a
  quotient can be obtained by finding the powers of
  numerator and denominator and dividing them.

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Properties

• 23. Zero Power Property
• 24. Negative Power Property

• 23. If a variable has an exponent of zero, then
  it must equal one 301
• 24. When a fraction or a number has negative
  exponents, you must change it to its reciprocal
  in order to turn the negative exponent into a
  positive exponent

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Properties

• 25. zero product property

• 25. when your variables are equal to zero then
  one or the other must be zero.

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In the next slides you will review

• Product of Roots Property
• Quotient of Roots Property
• Root of a Power Property
• Power of a Root Property

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Properties

• 26. Product of Roots Property

• 26. The product is the same as the product of
  square roots
• X

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Properties

• 27. Quotient of Roots Property

• 27. the quotient is the same as the quotient of
  the square roots

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Properties

• 28. Root of a Power Property
• 29. Power of a Root Property

• 28.
• 29.

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Property quiz

• Problems in which you determine the property.
• You will fill in the answer on the power point
• when finished go back through the properties to
  make sure you have the correct answers.
• 1.
• 3.
• 4.
• 5.

• A. if a b, then a c b c.
• B. aa
• C. If a b and b c, then a c.
• D. answer will still not chage.(3 x 2) x 4 24
  or 3 x (2 x 4) 24.
• E. 4 x 2 2 x 4

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Solving1st power equations

• In the next slides you will see how to-
• A. with only one inequality sign
• B. conjunction
• C. disjunction

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Solving1st power equations-with only one
inequality sign

• This will only be true if x is equal to four
• The answer will be x gt 4
• Which on a number line is

• 6x 24
• 6x gt 24
• x gt 4

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Solving1st power equations- conjunction

• A conjunction is true only if both the statements
  in it are true
• A conjunction is a mathematical operator that
  returns an output of true if and only if all of
  its operands are true.

• -2 lt x lt 4

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Solving1st power equations-disjunction

• A disjunction is statement which connects two
  other statements using the word or.
• To solve a disjunctions of two open sentences,
  you find the variables for which at least one of
  the sentences is true. The graph consists of all
  points that are in the graph

• Ex. -3ltx or xlt4
• Line where the lines

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Linear equations in two variables

• Standard form
• Next determine whether or not the equations is
  linear or not.
• Next subtract 5x from both sides

• Ax By C
•   y 5x - 3
• 5x y -3
• This would be -5x y -3 it would become a
  straight line

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Linear equations in two variables cont.

• A graphed linear equation

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Linear systems

• A. substitution
• B. addition/subtraction
• C. check for understanding of terms-
• 1.dependent
• 2. inconsistent
• 3. consistent

• Solving equations in two variables
• Graphing points
• Standard/General Form
• Slope- Intercept Form
• Point-Slope Form
• Slopes

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Linear systems-substitution

• 1.looks like it would be easy to solve for x, so
  we take it and isolate x
• 2. Now that we have y, we still need to
  substitute back in to get x. We could substitute
  back into any of the previous equations, but
  notice that equation 3 is already conveniently
  solved for x
• 3. answer is 1

• 1.2y  x  3
• 2. 2y  x  3
• 3.x3-2y
• x3-2(1)
• x3-2
• x1

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Linear systems-add/sub (elimination)

• 1. Note that, if I add down, the y's will cancel
  out. So I'll draw an "equals" bar under the
  system, and add down
• 2. Now I can divide through to solve for x 5,
  and then back-solve, using either of the original
  equations, to find the value of y. The first
  equation has smaller numbers, so I'll back-solve
  in that one

• 1. 2x y 9 3x y 16
• 2. 2x y 9 3x y 16
• 5x 25
• 3. 2(5) y 9  10 y 9          y 1

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Linear systems-understanding terms

• 1. inconsistent
• 2. consistent
• 3. dependent

• A system is inconsistent if it has no solutions
• A system is consistent if there is at least one
  solution
• A system is dependent if it has many solutions

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Factoring-methods and techniques

• A. Factoring GCF
• B. Difference of squares
• C. Sum and difference of cubes
• D. Reverse of foil
• E. PST
• F. Factoring by grouping

• In the next slides you would learn each.

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Factoring GCF

• EXAMPLE
• these are the steps you'll need to go through.
• 1.3x2 6x - 4x - 8
• 2. (3x2 6x) - (4x 8)
• 3 3x (x 2) - 4 (x 2)
• 4.(3x - 4) (x 2)

• grouping is important
• pulling out the GCF will take one or two times

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Difference of squares-binomials

• you must find out what is a common factor
• then make into binomials
• You must watch squares in case answer might be
  prime

• EXAMPLE
• 1.a2-b2
• 2.(ab)(a-b)a2-b2
• Prime example
• EXAMPLE
• 1.a2b2

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Sum and difference of cubes-binomials

• find difference
• opposite product in the middle
• Use parenthesis very important.

• EXAMPLE
• 1. x3 -8
• 2.x3 23
• 3. (x-2)(x22x22)
• 4.(x-2)(x22x4)

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Reverse of foil-trinomials

• Just do foil in reverse
• Trial and error it may take you a couple of tries
  to find the correct answer.

• EXAMPLE
• 1.3x2 - 6x x - 2
• 2.(3x1)(x-2)

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PST-perfect square trinomial

• The first term and the last term will be perfect
  squares.
• The coefficient of the middle term will be double
  the square root of the last term multiplied by
  the square root of the coefficient of the first
  term.
• There will be many different problems that will
  be PST

• EXAMPLE
• 1.x2 6x 9 0
• 2.x2 2(3)x 32 0
• 3.(x 3)2 0
• 4. x30
• 5.x-3
• EXAMPLE
• (ax)2 2abx b2

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Factoring by grouping-four or more items

• remember it is a binomial and make sure you set
  problem up for globs
• the key is to find a common factor and keep
  factoring out the problem

• EXAMPLE
• 1. x3-4x23x-12
• 2.x3-4x23x-12x2(x-4)3(x-4)
• 3.(x-4)(x23)

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Functions

• A Function is a correspondence between two sets,
  the domain and the range, that assigns to each
  member of the domain exactly one member of the
  range. Each member of the range must be assigned
  to at least one member of the domain.

example of equation h(k) x2 - 2x -2
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Simplifying expressions with exponents

• You would use properties when doing this.
• The x6 means six copies of x multiplied together
  and the x5 means five copies of x multiplied
  together. So if I multiply those two expressions
  together, I will get eleven copies of x
  multiplied together.

• x6 x5
• x6 x5 (x6)(x5)              
  (xxxxxx)(xxxxx)    (6 times, and then 5
  times)              xxxxxxxxxxx         (11
  times)               x11  

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Simplifying expressions with exponents cont.

• The exponent rules tell me to subtract the
  exponents. But let's suppose that I've forgotten
  the rules again. The " 68 " means I have eight
  copies of 6 on top the " 65 " means I have five
  copies of 6 underneath.
• Then you would cancel out the top and bottom then
  you would have your simplified expression.

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Word problems

• In three more years, Jack's grandmother will be
  six times as old as Jack was last year. If Jack's
  present age is added to his grandmother's present
  age, the total is 68. How old is each one now?

• Let 'g' be Jack's grandmother's current age
• Let 'j' be Jack's grandmother's current age
• If Jack's present age is added to his
  grandmother's present age, the total is 68
• j g 68
• In six more years, Jack's grandmother will be six
  times as old as Jack was last year
• (g3) 6 (j-1)
• If Jack's present age is added to his
  grandmother's present age, the total is 68
• jg68
• Solving both equations we get Jack's age (j) as
  11 and Jack's grandmother's age (g) as 57

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Lines best fit or regression

• A Regression line is a line draw through and
  scatter-plot of two variables. The line is
  chosen so that it comes as close to the points as
  possible.
• When asked to draw a linear regression line or
  best-fit line, you have to to draw a line through
  data point on a scatter plot. In order to solve
  these problems a calculator will be needed

• Lines best fit or regression

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Conclusion

• These slides should have gave you information on
  what we worked on during semester two and what
  you will have to know for the test.