Algebra 1

1
Algebra 1

• Marcos De la Cruz
• Algebra 1(6th period)
• Ms.Hardtke
• 5/14/10

2
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

3
Addition Property of Equality

• If the same number is added to both sides of an
  equation, both sides will be and remain equal
• 33 (equation)
• If 2 is added to both sides
• 2332
• 55

• Negative Special Case
• y3x5 (equation)
• If (-3) is added to both sides
• Y-33x5-3
• Y-33x2
• Its still equal

4
Multiplication Property of Equality

• States that when both sides of an equal equation
  is multiplied and the equation remains equal
• If 55 (equation)
• 5x22x5
• You multiply both sides by 2
• 1010
• Still remains equal

5
Reflexive Property of Equality

• When something is the exact same on both sides
• A A
• 7x 7x
• 3456x 3456x

6
Symmetric Property of Equality

• When two variables are different but are the same
  number/amount (equal symmetry)
• If ab, then ba
• If cd, then dc
• If xypxyo, then xyoxyp

7
Transitive Property of Equality

• When numbers or variables are all equal
• If ab and bc, then ca
• if 5x100 and 1004y, then 4y5x
• if 02x and 2x78p, then 78p0

8
Associative Property of Addition

• The sum of a set of numbers is the same no matter
  how the numbers are grouped. Associative property
  of addition can be summarized algebraically as
• (a b) c a (b c)
• (3 5) 2 8 2 10
• 3 (5 2) 3 7 10
• (3 5) 2 3 (5 2).

9
Associative Property of Multiplication

• The product of a set of numbers is the same no
  matter how the numbers are grouped. The
  associative property of multiplication can be
  summarized algebraically as
• (ab)c a(bc)

10
Commutative Property of Addition

• The sum of a group of numbers is the same
  regardless of the order in which the numbers are
  arranged. Algebraically, the commutative property
  of addition states
• a b b a
• 5 2 2 5 because 5 2 7 and 2 5 7
• -3 11 11 - 3

11
Commutative Property of Multiplication

• The product of a group of numbers is the same
  regardless of the order in which the numbers are
  arranged. Algebraically, commutative property of
  multiplication can be written as
• ab ba
• 8x6 48 and 6x8 48 thus, 8x6 6x8

12
Distributive Property

• The sum of two addends multiplied by a number is
  the sum of the product of each addend and the
  number
• A(bc)
• Ab Ac
• 3x(2y4)
• 6xy 12x

13
Property of Opposites/Inverse Property of Addition

• When a number is added by itself negative or
  positive to make zero
• a (-a) 0
• 5 (-5) 0
• -3y (3y) 0

14
Property Of Reciprocals/Inverse Property of
Multiplication

• For two ratios, if a/b c/d, then b/a d/c
• a(1/a) 1
• 5(1/5) 1
• 8/1 x 1/8 1

• A number times its reciprocal, always equals one
• A Reciprocal is its reverse and opposite (the
  signs switch from to or vice versa)

15
Reciprocal Function (continued)

• The reciprocal function y 1/x. For every x
  except 0, y represents its multiplicative inverse

16
Identity Property of Addition

• A number that can be added to any second number
  without changing the second number. Identity for
  addition is 0 (zero) since adding zero to any
  number will give the number itself
• 0 a a 0 a
• 0 (-3) (-3) 0 -3
• 0 5 5 0 5

17
Identity Property of Multiplication

• A number that can be multiplied by any second
  number without changing the second number.
  Identity for multiplication is "1, instead of 0,
  because multiplying any number by 1 will not
  change it.
• a x 1 1 x a a
• (-3) x 1 1 x (-3) -3
• 1 x 5 5 x 1 5

18
Multiplicative Property of Zero

• Anything number or variable multiplied times zero
  (0), will always equal zero
• 5 x 00
• 5g x 00
• No matter what number is being multiplied by
  zero, it will always be zero

A really long way to explain the Multiplicative
Property of Zero (Proof)
http//upload.wikimedia.org/math/b/5/8/b5892630f1d
2f28a580331a1d7e3e79f.png
19
Closure Property of Addition

• Sum (or difference) of 2 real numbers equals a
  real number
• 10 (5) 5

20
Closure Property of Multiplication

• Product (or quotient if denominator 0) of 2 Reals
  equals a real number
• 5 x 2 10

21
(Exponents) Product of Powers Property

• Exponents
• Exponents are the little numbers above numbers,
  that mean that the number is multiplied by itself
  that many times
• 7 7
• (7 7) (7 7 7 7 7 7)

• When two exponents or numbers with exponents are
  being multiplied, you add both exponents, but you
  still multiply the number or variable
• 3x (5x )
• 15x
• 15x

3
4
2
6
(34)
7
22
Power of a Product Property

• To find a power of a product, find the power of
  each factor and then multiply.  In general
• (ab) a b
• Or
•  a b (ab)
• (3t)
• (3t) 3 t 81t

m
m
m
m
m
m
4
4
4
4
4
23
Power of a Power Property

• To find a power of a power, multiply the
  exponents. (Its basically the same as the Power
  of a Product Property, if forgotten, go one slide
  back and review.)
• (5 )
• (5 )(5 )(5 )(5 ) 5 5
• Its basically this
• (a ) a

3
4
3
3
3
3
3(4)
12
b
c
bc
24
Quotient of Powers Property
b
c
b-c

• When both the denominator and numerator of a
  fraction have a common variable, it can be
  canceled, therefore not usable anymore
• Also when a variable is canceled, the exponents
  are subtracted, instead of added as in the
  Product of Powers Property

• a /a a
• 5 5x5x5
• 
  
• 5 5x5
• 5
• (the canceling of common factors)

3
2
25
Power of a Quotient Property

• This is almost the same as the Quotient of Powers
  Property, but this time, an entire fraction is
  multiplied by an exponent
• You also have to cancel the common factors, if
  there are any
• (a/b) a /b (a/6)
• (and vice versa) (a /36)

c
c
c
2
2
26
Zero Power Property

• If a variable has an exponent of zero, then it
  must equal one
• a 1
• b 1
• c b a 1
• (a ) 1

0
0
0
0
0
2
0
27
Negative Power Property

• 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
• 4 ¼ 1/16

-2
2
The exponent turned from negative to positive
28
Zero Product Property

• When both variables equal zero, then one or the
  other must equal zero
• if ab0, then either a0 or b0
• if xy0, then either x0 or y0
• if abc0, then either a0, b0, or c0

29
Product of Roots Property

• The product is the same as the product of square
  roots

30
Quotient of Roots Property

• The square root of the quotient is the same as
  the quotient of the square roots

A
A
B
B
31
Root of a Power Property
32
Power of a Root Property
33
Density Property of Rational Numbers

• Between any two rational numbers, there exists at
  least one additional rational number

1 2 3 4
5 6 7 8
9
4.5 or 4½
34
Websites

• PROPERTIES
• http//www.my-ice.com/ClassroomResponse/1-6.htm
• http//intermath.coe.uga.edu/dictnary/descript.asp
  ?termID300
• http//en.wikipedia.org/wiki/Multiplicativeinverse
  
• http//ask.reference.com/related/ReciprocalofaN
  umber?qsrc2892ldiro10601
• http//faculty.muhs.edu/hardtke/Alg1_Assignments.h
  tm MUHS
• http//www.northstarmath.com/sitemap/Multiplicativ
  eProperty.html
• http//hotmath.com/hotmath_help/topics/product-of-
  powers-property.html
• http//hotmath.com/hotmath_help/topics/power-of-a-
  product-property.html
• http//hotmath.com/hotmath_help/topics/power-of-a-
  power-property.html
• http//hotmath.com/hotmath_help/topics/quotient-of
  -powers-property.html
• http//hotmath.com/hotmath_help/topics/power-of-a-
  quotient-property.html
• http//hotmath.com/hotmath_help/topics/properties-
  of-square-roots.html
• http//www.slideshare.net/misterlamb/notes-61
• http//www.ecalc.com/math-help/worksheet/algebra-h
  elp/

Hotmath.com
35
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

36
Standard/General Form

• Standard Form
• Ax By C
• The terms A, B, and C are integers (could be
  either positive or negative numbers or fractions)
• If Fractions
• Multiply each term in the equation by its LCD
  (Lowest Common Denominator)
• Either add or subtract to get either X or Y
  isolated, in one side of the

• If Decimals
• Multiply each term in the equation depending on
  the decimal with the most numbers (by 10, 100,
  1000, etc)
• 1.23 (multiply times 100)
• 123.00
• Subtract or add to get X or Y isolated
• If Normal Numbers (neither fractions or
  decimals)
• Just add or subtract to get X or Y isolated

37
Graph Points

• A Graph Point contains of an X and a Y
• (x,y)
• The X and Y mean where exactly the point is
  located

Y line graph
X line graph
38
Standard/General Form Ex.

• Fractions

• You multiply by the LCM
• Which in this case is 20x
• Then to double check it

39
Point-Slope Form

• The Y on the Point-Slope form., doesnt mean
  that the Y is multiplied by one, but it means to
  use the first Y of the two or one point given as
  a problem (same with X)
• (4,3) and the slope is 2
• M slope
• Ystays the same
• X is 4 (because 4 is in the x spot)
• Y is 3
• Xstays the same
• If the problem gives you two points and no slope,
  then you are free to choose what which or the Xs
  or the Ys you may want to use for your
  Point-Slope Form.

• The Point-Slope form. got its name because it
  uses a single point in a graph and a on the slope
  of the line
• It is usually used to find the slope of a graph,
  if the slope is not given in a certain problem or
  equation

1
ex
1
1
40
Point-Slope Form

• (4,3) and m2
• you must convert it to a slope-intercept form
• YMx B
• Y-3 2(x-4)
• Y-3 2x-8
• Y 2x 11 (slope-intercept form)

41
Slope-Intercept Explanation

• ymxb
• Sometimes in the Slope-intercept form, there are
  fractions as the slope or the y-intercept
• B y-intercept
• Rise/Run
• When the slope is a fraction, you mark the B in a
  graph, which is the y-intercept
• Then depending on the slope, if its positive than
  the line will look like this
• If its not positive, but negative, it will look
  like

42
Point-Slope (Slope-Intercept) Graph

• Y 2x 11

Rise/Run Go up twice and to the side once
(5,0)
(0,-11)
43
Websites(for further information)

• Linear Equations
• http//www.algebralab.org/studyaids/studyaid.aspx?
  fileAlgebra1_5-5.xml
• http//www.freemathhelp.com/point-slope.html
• http//www.wonderhowto.com/how-to-solve-mixed-equa
  tion-decimal-percent-fraction-303082/

44
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

45
Linear Systems Method Explanation

• Substitution
• The Substitution Method, is used when, there are
  two equations, and you pick one (the one that
  looks the easiest to do) and you isolate either
  the x or the y
• When x or y is isolated, then you will get
  something like this
• Y ?x ?
• X ?y ?
• Then, you replace the x or the y in the equation
  that you didnt touch yet, and you must insert
• If you isolated the y, then you will solve for x
• If you isolated the x, then you will solve for y

• Elimination
• The Elimination Method, is used when there are
  two equations and, it is said to be a lot easier
  than the Substitution Method
• First, you will have to decide whether you want
  to go for the x or the y
• Then, you will multiply and cancel/eliminate
  either x or y depending, on which one did you
  chose to do (x or y)
• Then you solve for x or y
• You will eventually substitute, more like insert
  your y or x answer into the either problem
  replacing it with x or y
• Then you solve for either x or y

46
Substitution Method
y 11 - 4x
Isolate the Y or X
Substitute the number, insert it
x 2(11 - 4x) 8
Solve for X and Solve for Y (vice versa)
Answers
47
Literal Coefficients
http//www.tpub.com/math1/13d.htm
Simultaneous equations with literal coefficients
and literal constants may be solved for the value
of the variables just as the other equations
discussed in this chapter, with the exception
that the solution will contain literal numbers.
For example, find the solution of the system
We proceed as with any other simultaneous linear
equation. Using the addition method, we may
proceed as follows To eliminate the y term we
multiply the first equation by 3 and the second
equation by -4. The equations then become
                      
To eliminate x, we multiply the first equation by
4 and the second equation by -3. The equations
then become                       We
may check in the same manner as that used for
other equations, by substituting these values in
the original equations.
3 Variables !!
48
Elimination Method

• 2x 3y 19
• 5x 2y 20
• 2x 3y 19 (2)
• 5x 2y 20 (-3)
• 4x 6y 38
• -15x 6y -60
• -11x -22
• X 2
• 2x 3y 19
• 2(2) 3y 19
• 4 3y 19
• -3y 15
• Y -5

• The two equations
• Now we multiply and then later cancel out a
  variable, depending which one you chose
• Now we got one answerx 2
• Now we must insert the two, into the either of
  the equations(substitution method)
• Now you got the y -5

49
Dependent

• When a system is "dependent," it means that ALL
  points that work in one of them ALSO work in the
  other one
• Graphically, this means that one line is lying
  entirely on top of the other one, so that if you
  graphed both, you would really see only one line
  on the graph, since they are imposed on top of
  each other
• One of them totally DEPENDS on the other one

50
Independent

• When a system is "independent," it means that
  they are not lying on top of each other
• There is EXACTLY ONE solution, and it is the
  point of intersection of the two lines
• It's as if that one point is "independent" of the
  others.
• To sum up, a dependent system has INFINITELY MANY
  solutions. An independent system has EXACTLY ONE
  solution

51
Consistent

• We say that a point is a "solution" to the system
  when it makes BOTH equations true, right?
• This is to say that there exists a point (or set
  of points) that "work" in one equation and also
  "work" in the other one
• So we say that this point is CONSISTENT from one
  equation to the next

52
Inconsistent

• On the other hand, if there are NO points that
  work in both, then we say that the equations are
  INCONSISTENT
• NO numbers that work in one are consistent with
  the other
• To sum up, a consistent system has at least one
  solution. An inconsistent system has NO solution
  at all

53
Websites

• Linear Systems
• http//www.tpub.com/math1/13d.htm
• http//mathforum.org/library/drmath/view/62538.htm
  l
• http//www.purplemath.com/modules/systlin2.htm

54
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

55
1st Power Equations (1 Variable)

• In order to get the answer, when there is only
  one variable
• You must, isolate the variable, and if it has a
  sign with it (a negative sign) or a number with
  it, than you can and must divide the number to
  the other side
• In order to get the variable completely alone
• Then you get your answer

56
1 Variable Problems

• 5x 3 2 (2 3x)
• 5x 3 4 6x
• 5x 4 (-3) 6x
• 5x 1 6x
• 5x 6x 1
• 11x 1
• X 1/11
• 2x 8
• X 4

• -x 20 3x 2(5x 10)
• -x 20 3x 10x 20
• -x 7x 40
• -8x -40
• X 5
• These 1 variable problems are fairly simple and
  easy
• All you have to do is isolate the variable
• Then just add, subtract, or divide and solve the
  problem

57
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

58
Factoring FOIL

• FOIL
• is a type of factoring that includes two globs
• (3x 2)(3x 2)
• this FOIL means that the O and I in FOIL, will
  be the same number, but one will be negative and
  one positive, therefore, they will cancel each
  other out

59
PST

• PST, is when two globs are reversed FOILed and
  they equal perfectly
• (x 2)(x 2)
• X 4x 4 (PST)
• The first number in a PST, to check if you got a
  PST, the first number has to be squared, and if
  its not, then take out the GCF
• The First and Last number should have roots,
  while the middle number should be the double of
  the roots of both the First and Last number

2
60
Factor GCF

• The GCF stands for the Greatest Common Factor
• Which means, that if you have a binomial or a
  trinomial with prime numbers in common or more
  variables than needed, then you can factor them
  out, and then continue to solve the problem
• Whatever you factored out, will still be part of
  the Answer of the problem

61
Difference of Squares

• First take out the GCF (always)
• If there are two globs that if FOILed, arent a
  PST, but they just make a binomial, but it can be
  divided into two more binomials
• Then you have conjugates
• (? ?) (? - ?)
• As long as you have a negative glob, that you can
  still divide into more globs, you can continue to
  divide, but if one glob is the same as another
  glob, then your answer will only contain the
  glob, but only once

62
Sum or Difference of Cubes

• The Sum or Difference of Cubes, is when you take
  variable squares or numbers with roots cubed, and
  they are separated and into a binomial and a
  trinomial

63
Reverse FOIL

• This is the same thing as FOIL factoring, but
  there is a Trial and Error system
• That means, that when given trinomial, you will
  have to guess and check if it FOILs the correct
  globs, and you will have to continue to do that,
  until you get the correct globs

64
Factor By Grouping

• 4x4
• It is a binomial because there are two terms, and
  a repeated glob, it is a common glob
• Which means GCF
• 2x2
• Sometimes you can rearrange the order of the
  terms, to find the correct glob

65
Factor By Grouping

• 3x1
• Rearrange into a PST
• Then make two perfect globs
• If conjugates then separate them

66
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

67
Rational Expressions

• PST
• X 10x 25
• (x 5)

• We define a Rational Expression as a fraction
  where the numerator and the denominator are
  polynomials in one or more variables.

2
2
Addition and Subtraction of Rational Expressions
        3           5              
8               (2)(4)            2            
                                           
                     20         20            
20             (4)(5)          5
Multiplication and Division of Rational
Expressions
2
  3x - 4x              x(3x - 4)             3x -
4                                           
                  2x - x                x(2x -
1)             2x - 1
2
68
R.E
FOIL
First - multiply the first term in each set of
parenthesis 4x x 4x2                      
                                                  
       Outside - multiply the two terms on the
outside 4x 2 8x                           
                                               
Inside - multiply both of the inside terms 6
x 6x                                         
                                  Last -
multiply the last term in each set of
parenthesis 6 x 2 12                        
                                                  
 
69
Websites

• Rational Expressions
• http//www.freemathhelp.com/using-foil.html

70
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

71
Completing the Square (1)

• (X n)²   X² 2nx n²
• Note the rightmost term (n²) is relatedto 2n
  (the x coefficient) by the formula
• Solving this by "completing the square" is as
  follows1) Move the "non X" term to the right
• 4X² 12X 16
• 2) Divide the equation by the coefficient of X²
  which in this case is 4
• X² 3X 4
• 3) Now here's the "completing the square" stage
  in which we
•       take the coefficient of X      divide
  it by 2      square that number      then
  add it to both sides of the        equation.

72
Completing the Square (2)

• In our sample problem       the coefficient of X
  is 3      dividing this by 2 equals 1.5     
  squaring this number equals (1.5)² 2.25     
  Now, adding that to both sides of the equation,
  we have
• X² 3X 2.25 4 2.25
• 4) Finally, we can take the square root of both
  sides of the equation and we have
• X 1.5 Square Root (4 2.25)
• X Square Root (6.25) -1.5
• X 2.5 -1.5
• X 1.0
• Let's not forget that the other square root of
  6.25 is -2.5 and so the other root of the
  equation is
• (-2.5 -1.5) -4

73
Quadratic Formula
We can follow precisely the same procedure as
above to derive the Quadratic Formula.   All
Quadratic Equations have the general form aX²
bX c 0                                    
                                                  
                                                  
                                                  
                                                  
            
74
Discriminant and the Quadratic Equation

• The Discriminant is a number that can be
  calculated from any quadratic equation A
  quadratic equation is an equation that can be
  written as
• ax ² bx c where a ? 0
• The Discriminant in a quadratic equation is found
  by the following formula and the discriminant
  provides critical information regarding the
  nature of the roots/solutions of any quadratic
  equation. discriminant b² - 4acExample of the
  discriminant
• Quadratic equation y 3x² 9x 5
• The discriminant 9 ² - 4 3 5

75
Quadratic Equation

• Quadratic Equation y x² 2x 1
• a 1
• b 2
• c 1
• The discriminant for this equation is 2² - 41
  1 4 - 4 0 Since the discriminant of zero,
  there should be 1 real solution to this equation.
  Below is a picture representing the graph and one
  solution of this quadratic equation Graph of y
  x² 2x 1                                        
               

76
Websites

• R.E.
• http//www.mathwarehouse.com/quadratic/discriminan
  t-in-quadratic-equation.php
• http//webgraphing.com/quadraticequation_quadratic
  formula.jsp
• http//webmath.com/quadtri.html

77
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

78
F(x)

• In Algebra f(x) is another symbol for y
• Y 3
• F(x) 3
• Its practically the same things, but people use
  it for confusion

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Domain and Range

• Domain
• For a function f defined by an expression with
  variable x, the implied domain of f is the set of
  all real numbers variable x can take such that
  the expression defining the function is real. The
  domain can also be given explicitly.
• Range
• The range of f is the set of all values that the
  function takes when x takes values in the domain

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Domain
Example The function y v(x 4) has the
following graph

• The domain of the function is x -4, since x
  cannot take values less than -4. (Try some values
  in your calculator, some less than -4 and some
  more than -4. The only ones that "work" and give
  us an answer are the ones greater than or equal
  to -4).
• Note
• The enclosed (colored-in) circle on the point
  (-4, 0). This indicates that the domain "starts"
  at this point.
• That x can take any positive value in this
  example

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Range
Example 1 Let's return to the example above, y
v(x 4). We notice that there are only positive
y-values. There is no value of x that we can find
such that we will get a negative value of y. We
say that the range for this function is y 0.
Example 2 The curve of y sin x shows the range
to be betweeen -1 and 1
The domain of the function y sin x is "all
values of x", since there are no restrictions on
the values for x.
http//www.intmath.com/Functions-and-graphs/2a_Dom
ain-and-range.php
82
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

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Solving Inequalities

• Linear inequalities are also called first degree
  inequalities, as the highest power of the
  variable in these inequalities is 1. E.g.  4x gt
  20 is an inequality of the first degree, which is
  often called a linear inequality.
• Many problems can be solved using linear
  inequalities.
• We know that a linear equation with one
  pronumeral has only one value for the solution
  that holds true. For example, the linear equation
  6x 24 is a true statement only when x 4.
  However, the linear inequality 6x gt 24 is
  satisfied when x gt 4. So, there are many values
  of x which will satisfy the inequality 6x gt 24.

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Inequalities

• Recall that
• the same number can be subtracted from both sides
  of an inequality
• the same number can be added to both sides of an
  inequality
• both sides of an inequality can be multiplied (or
  divided) by the same positive number
• if an inequality is multiplied (or divided) by
  the same negative number, then

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Inequalities
                                                  
                                                  
    
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Conjunctions

• When two inequalities are joined by the word and
  or the word or, a compound inequality is formed.
  A compound inequality like-3 lt 2x 5          
  and 2x 5 7is called a conjunction, because
  it uses the word and. The sentence -3 lt 2x 5
  7 is an abbreviation for the preceding
  conjunction. Compound inequalities can be solved
  using the addition and multiplication principles
  for inequalities.

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Disjunction

• A compound inequality like 2x - 5 -7 or is
  called a disjunction, because it contains the
  word or. Unlike some conjunctions, it cannot be
  abbreviated that is, it cannot be written
  without the word or.

88
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

89
Word Problems

• The sum of twice a number plus 13 is 75. Find the
  number.
• The word is means equals. The word and means
  plus. Therefore, you can rewrite the problem like
  the following
• The sum of twice a number and 13 equals 75.
• Using numbers and a variable that represents
  something, N in this case (for number), you can
  write an equation that means the same thing as
  the original problem. 2N 13 75
• Solve this equation by isolating the variable.
  2N 13 75 Equation. - 13 -13 Add (-13) to
  both sides. ------------- 2N 62
• N 31
• Divided both sides by 2

90
Word Problems

• 2) Find a number which decreased by 18 is 5 times
  its opposite.

Again, you look for words that describe equal
quantities. Is means equals, and decreased by
means minus. Also, opposite always means
negative. Keeping that information in mind makes
it so an equation can be written that describes
the problem, just like the following N - 18
5(-N) Equation. N - 18 -5N Multiplied out. 5N
18 5N 18 Add (5N 18) to ------------------
both sides. 6N 18 N 3 Divide both sides by 6
to isolate N.
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Word Problems

• 3) Julie has 50, which is eight dollars more
  than twice what John has.  How much has John?
  First, what will you let x represent?
• The unknown number -- which is how much that John
  has.
• What is the equation?
• 2x 8 50.
• Here is the solution
• x 21

92
Word Problems

• 4) Carlotta spent 35 at the market.  This was
  seven dollars less than three times what she
  spent at the bookstore how much did she spend
  there?
• Here is the equation.
• 3x - 7 35
• Here is the solution
• x 14

93
Algebra Topics

• 1- Properties
• 2- Linear Equations
• 3- Linear Systems
• 4- Solving 1st Power Equations (1 Variable)
• 5- Factoring
• 6- Rational Expressions
• 7- Quadratic Equations
• 8- Functions
• 9- Solving 1st Power Inequalities (1 Variable)
• 10- Word Problems
• 11- Extras

94
11 - The END