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

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

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

If the same number is added to both sides of an equation, both sides will be and remain equal 3=3 (equation) If 2 is added to both sides 2+3=3+2 5=5 Negative Special Case y=3x+5 (equation) If (-3) is added to both sides Y-3=3x+5-3 Y-3=3x+2 Its still equal

Multiplication Property of Equality
States that when both sides of an equal equation is multiplied and the equation remains equal If 5=5 (equation) 5x2=2x5 You multiply both sides by 2 10=10 Still remains equal

Reflexive Property of Equality
When something is the exact same on both sides A = A 7x = 7x 3456x = 3456x

Symmetric Property of Equality
When two variables are different but are the same number/amount (equal symmetry) If a=b, then b=a If c=d, then d=c If xyp=xyo, then xyo=xyp

Transitive Property of Equality
When numbers or variables are all equal If a=b and b=c, then c=a if 5x=100 and 100=4y, then 4y=5x if 0=2x and 2x=78p, then 78p=0

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 = = 10 3 + (5 + 2) = = 10 (3 + 5) + 2 = 3 + (5 + 2).

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)

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 = because = 7 and = 7 =

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

Distributive Property
The sum of two addends multiplied by a number is the sum of the product of each addend and the number A(b+c) Ab + Ac 3x(2y+4) 6xy + 12x

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

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)

Reciprocal Function (continued)
The reciprocal function: y = 1⁄x. For every x except 0, y represents its multiplicative inverse

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

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

Multiplicative Property of Zero
Anything number or variable multiplied times zero (0), will always equal zero 5 x 0=0 5g x 0=0 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)

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

Closure Property of Multiplication
Product (or quotient if denominator 0) of 2 Reals equals a real number 5 x 2 = 10

(Exponents) Product of Powers Property
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 3 4 2 6 (3+4) 7

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

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 ) = Its basically this: (a ) = a 3 4 3 3 3 3 3(4) 12 b c bc

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 x5x5 —— —————— x5 = 5 (the canceling of common factors) 3 2

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

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 2

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 ¼ /16 -2 2 The exponent turned from negative to positive

Zero Product Property When both variables equal zero, then one or the other must equal zero if ab=0, then either a=0 or b=0 if xy=0, then either x=0 or y=0 if abc=0, then either a=0, b=0, or c=0

Product of Roots Property
The product is the same as the product of square roots

Quotient of Roots Property
The square root of the quotient is the same as the quotient of the square roots: A A B B

Root of a Power Property

Power of a Root Property

Density Property of Rational Numbers
Between any two rational numbers, there exists at least one additional rational number 4.5 or

Websites PROPERTIES Hotmath.com
— MUHS Hotmath.com

Algebra Topics 2- Linear Equations 1- Properties 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

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

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

Standard/General Form Ex.
Fractions: You multiply by the LCM Which in this case is 20x Then to double check it…

Point-Slope Form The Y on the Point-Slope form., doesn’t 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 Y—stays the same X —is 4 (because 4 is in the x spot) Y — is 3 X—stays 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

Point-Slope Form (4,3) and m=2 Y=Mx + B
you must convert “it” to a slope-intercept form Y=Mx + B Y-3 = 2(x-4) Y-3 = 2x-8 Y = 2x – 11 (slope-intercept form)

Slope-Intercept Explanation
y=mx+b 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:

Point-Slope (Slope-Intercept) Graph
Y = 2x – 11 Rise/Run Go up twice and to the side once (5,0) (0,-11)

Websites (for further information)
Linear Equations

Algebra Topics 3- Linear Systems 1- Properties 2- Linear Equations
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

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 didn’t 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

Substitution Method Isolate the Y or X
y = x 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

Literal Coefficients 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 !!

Elimination Method 2x – 3y = 19 5x – 2y = 20 The two equations
Now we multiply and then later cancel out a variable, depending which one you chose Now we got one answer—x = 2 Now we must insert the two, into the either of the equations…(substitution method) Now you got the y = -5

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

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

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

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

Websites Linear Systems: http://www.tpub.com/math1/13d.htm

Algebra Topics 4- Solving 1st Power Equations (1 Variable)
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

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

1 Variable Problems 5x + 3 = 2 (2 – 3x) 2x = 8 5x + 3 = 4 – 6x
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

Algebra Topics 5- Factoring 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

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

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

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

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

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

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

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

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

Algebra Topics 6- Rational Expressions 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

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

R.E Outside - multiply the two terms on the outside: 4x * 2 = 8x
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

Rational Expressions:
Websites Rational Expressions:

Algebra Topics 7- Quadratic Equations 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

Completing the Square (1)
(X + n)² =   X² + 2nx + n² Note the rightmost term (n²) is related to 2n (the x coefficient) by the formula Solving this by "completing the square" is as follows: 1) 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.

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 = 4) Finally, we can take the square root of both sides of the equation and we have: X = Square Root ( ) X = Square Root (6.25) -1.5 X = 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: ( ) = -4

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

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 − 4ac Example of the discriminant Quadratic equation = y = 3x² + 9x + 5 The discriminant = 9 ² − 4 • 3 •5

Quadratic Equation Quadratic Equation: y = x² + 2x + 1 a = 1 b = 2
The discriminant for this equation is 2² - 4•1 •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

Websites R.E.:

Algebra Topics 8- Functions 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

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

Domain and Range Domain Range
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

Domain Example: The function y = √(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

Range Example 1: Let's return to the example above, y = √(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.

Algebra Topics 9- Solving 1st Power Inequalities (1 Variable)
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

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 > 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 > 24 is satisfied when x > 4. So, there are many values of x which will satisfy the inequality 6x > 24.

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:

Inequalities

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

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.

Algebra Topics 10- Word Problems 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

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 Add (-13) to both sides N = 62 N = 31 Divided both sides by 2

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 N + 18 Add (5N + 18) to both sides. 6N = 18 N = 3 Divide both sides by 6 to isolate N.

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

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

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

11 - The END