Reviewing Can’t Hurt By: James Christian and Jack Gryniewicz

Reviewing Reviewing Can’t Hurt Can’t Hurt

By: James ChristianBy: James Christian

and Jack Gryniewiczand Jack Gryniewicz

Solving 1Solving 1stst Power Equations in Power Equations in One VariableOne Variable

Example problem: 5(4+n) = 2(9+2n)Example problem: 5(4+n) = 2(9+2n) 5(4+n) = 2(9=2n) Distribute the 5 and the 25(4+n) = 2(9=2n) Distribute the 5 and the 2 20+5n = 18+4n Subtract 4n from both sides20+5n = 18+4n Subtract 4n from both sides 20+n = 18 Subtract the 20 from both sides to get n alone 20+n = 18 Subtract the 20 from both sides to get n alone n = -2n = -2

Solving 1Solving 1stst Power Equations in Power Equations in One Variables (Cont’)One Variables (Cont’)

1 1

2 2 2

x

Fractional Coefficients- Example:

1st: get rid of denominator by multiplying by 2

2nd: divide by 1 on both sides

YOUR ANSWER IS 1

1 1x 1x


Variables in the Variables in the Denominator-Example:Denominator-Example:

1.1. Multiply by the LCD on Multiply by the LCD on both sidesboth sides

9(3-x) 9(3-x)

2.2. DistributeDistribute

3.3. Solve for xSolve for x

18 9 12 4x x 2 4

3 9

x

x

9(2 ) (3 )(4)x x

6

5x


Special Cases Special Cases 1.1. Variables cancels and you get the same number on both Variables cancels and you get the same number on both

sides of the equal sign sides of the equal sign

ALL ALL REALS REALS

2.2. Variables cancel but the numbers on either side of the equal Variables cancel but the numbers on either side of the equal sign are not the same sign are not the same

5 5x x 5 5

5 2x x 5 2

Properties

Addition Property of Equality If the same number is added to both sides of an equation, the

two sides remain equal. That is, if x = y, then x + z = y + z Multiplication Property of Equality

If a = b then a·c = b·c

Reflexive Property of Equality If something is equal to its identical twin a=a

Transitive Property of Equality If a = b, c = b so a = c

Symmetric Property of Equality If something flipped sides of the equal sign

So if a = b, then b = a

Properties

Distributive Property The sum of two numbers times a third number is equal to

the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3

Inverse Property of Addition The sum of a number and its additive inverse is always

zero. (x + (-x) = 0)

Closure Property of Addition Sum (or difference) of 2 real numbers equals a real

number

Properties

Commutative Property of Addition When two numbers are added, the sum is the same regardless of

the order of the addends. For example 4 + 2 = 2 + 4

Associative Property of Addition When three or more numbers are added, the sum is the same

regardless of the grouping of the addends. For example (2 + 3) + 4 = 2 + (3 + 4)

Additive Identity Property The sum of any number and zero is the original number. For

example 5 + 0 = 5

Properties

Commutative Property of Multiplication When two numbers are multiplied together, the product is the same

regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4

Associative Property of Multiplication When three or more numbers are multiplied, the product is the same

regardless of the grouping of the factors. For example (2 * 3) * 4 = 2

* (3 * 4) Identity Property of Multiplication

When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4)

Properties

Multiplicative Identity Property The product of any number and one is that number. For example 5 *

1 = 5.

Multiplicative Inverse Property any rational number times its reciprocal equals 1.

Multiplicative Property of Zero The product of 0 and any number results in 0.That is, for any real

number a, a × 0 = 0. Closure Property of Multiplication

For any two whole numbers a and b, their product a*b is also a whole number.Example: 10*9 = 90

Properties

Product of Powers Property This property states that to multiply powers having the

same base, add the exponents.

Power of a Product Property This property states that the power of a product can be

obtained by finding the powers of each factor and multiplying them.

Power of a Power Property This property states that the power of a power can be

found by multiplying the exponents.

Properties

Quotient of Powers Property This property states that to divide powers having the

same base, subtract the exponents.

Power of a Quotient Property This property states that the power of a quotient can be

obtained by finding the powers of numerator and denominator and dividing them.

Zero Power Property If the power is zero then the number will turn into 1.

Properties

Negative Power Property A negative exponent just means that the base is on the

wrong side of the fraction line, so you need to flip the base to the other side.

The Zero Product Property simply states that if ab = 0, then either a = 0 or b = 0

(or both). A product of factors is zero if and only if one or more of the factors is zero.

Properties

Product of Roots Property If you multiply two roots together you get the product.

Quotient of Roots Property The square root of a quotient is equal to the quotient

of the square root of the numerator and the square root of the denominator.

= ----

5 9 45

/a b a

b

Properties

Root of a Power Property The squared and the square root cancel each other out.

Power of a Root Property The square root and the square cancel each other out.

Power of a Root Property

2a a

2 5 5

Quiz Time!!! Lets see what you have learned.

A. Addition Property (of Equality)

B. Multiplication Property (of Equality) C. Reflexive Property (of Equality) D. Symmetric Property (of Equality) E. Transitive Property (of Equality) F. Associative Property of Addition G. Associative Property of Multiplication H. Commutative Property of Addition I. Commutative Property of Multiplication J. Distributive Property K. Prop of Opposites or Inverse Property of Addition L. Prop of Reciprocals or Inverse Property of Multiplication M. Identity Property of Addition N. Identity Property of Multiplication O. Multiplicative Property of Zero P. Closure Property of Addition Q. Closure Property of Multiplication R. Product of Powers Property S. Power of a Product Property T. Power of a Power Property U. Quotient of Powers Property V. Power of a Quotient Property W. Zero Power Property X. Negative Power Property Y. Zero Product Property Z. Product of Roots Property AA. Quotient of Roots Property BB. Root of a Power Property CC. Power of a Root Property

1. x9 ● x3 = _________2. = ________3. If x3 = y9, then y9 = _________4. 9(x – y) = ____5. (x9)3 = ____6. (xy)3 = ____7. If x3 = y9 and y9 = z12, then _________8. =_________9. (– 9xy)0 ____10. x3 ● ________ = x3 11.

12. x3 ● y9 = y9 ● ____

3x_______ 1

y

ANSWERS1. R

2. V

3. D

4. J

5. T

6. S

7. E

8. X

9. W

10. N

11. L

12. I

Inequality RulesInequality Rules

1.1. When the sign has the equal to option the When the sign has the equal to option the graph will have a closed circle and when the graph will have a closed circle and when the sign is only greater than or less than the sign is only greater than or less than the graph has an open circle.graph has an open circle.

2.2. DisjunctionDisjunction is an “or” statement when both is an “or” statement when both statements can be true. Satisfies one or both statements can be true. Satisfies one or both statements.statements.

3.3. A A conjunctionconjunction is when both statements are is when both statements are true and the statement has an “and” in it.true and the statement has an “and” in it.

Special Cases of InequalitiesSpecial Cases of Inequalities

Watch for no solution- conjunction does Watch for no solution- conjunction does not combine, but rather looks like a not combine, but rather looks like a disjunction when graphed.disjunction when graphed.

Watch for when every number works- Watch for when every number works- disjunction looks like a conjunction when disjunction looks like a conjunction when graphed.graphed.

One arrow- disjunction when one arrow One arrow- disjunction when one arrow proves the other to be right as well.proves the other to be right as well.

Solving Inequalities in One Solving Inequalities in One VariableVariable Don’t forget the Multiplication Don’t forget the Multiplication

Property of Inequality! If you Property of Inequality! If you multiply or divide by a negative, multiply or divide by a negative, the sign must be reversed.the sign must be reversed.

-5x 10-5x 10

x -2x -2 SIGN REVERSAL!!!

-2

Solution Set: Solution Set: {x: x > -2}{x: x > -2}

Try Solving These Inequalities Try Solving These Inequalities on Your Ownon Your Own

Disjunction- 5x > 25 or 3x < 9Disjunction- 5x > 25 or 3x < 9

Conjunction- -3x < 33 and x < 2Conjunction- -3x < 33 and x < 2

Linear Equations in Two Variables Slope:

(“m” stands for the slope) If you are given the points (3, –2) (9, 2)

Your slope would be:

Standard Form: Ax + By = C A, B, C are integers (positive or negative whole numbers) No fractions nor decimals in standard form. Traditionally the "Ax" term is positive.

Point-Slope Form: y – y1 = m(x – x1) For this one, they give you a point (x1, y1) and a

slope m, and have you plug it into this formula: y – y1 = m(x – x1)

1 2

1 2

y ym

x x

4 2

6 3

Graphing

Given the equations: y=x+1 y=2x

What would your intersecting point be???? The point (1,2) is where the two lines

intersect.

Linear SystemsLinear Systems

How many points do these How many points do these lines have in common?lines have in common?

Think about:Think about:

2x + y = 5 or y = - 2x + 52x + y = 5 or y = - 2x + 5

How to Find the Common How to Find the Common Points in Linear SystemsPoints in Linear Systems Methods-Methods-

1.1. Substitution- goal Substitution- goal is to get one is to get one variable equal to variable equal to an equation and an equation and substitute that substitute that expression into the expression into the other equation for other equation for that variable.that variable.

3x y 1

7x 2y 4

y

7x 2( ) 4

7x 6x 2 4

x 2

3x 1

3x 1

Arewedone

4

x 2 ?

How to Find the Common How to Find the Common Points in Linear SystemsPoints in Linear Systems Method 2-Method 2-

Estimate the SOLUTION Estimate the SOLUTION of a of a SYSTEMSYSTEM on a graph. on a graph.

Where do they intersect? Where do they intersect? 5

4

3

2

1

2 4

g x = -2x+5 f x = 3

2 x-2

y 2x 5

3y x 2

2

THE SOLUTION IS :

(2,1)

How to Find the Common How to Find the Common Point of Linear SystemsPoint of Linear Systems

Method 3- Elimination or Method 3- Elimination or Addition/Subtraction Addition/Subtraction -Goal is to combine equations in order -Goal is to combine equations in order to cancel one variableto cancel one variable

Steps to solving:Steps to solving:

1.1. Find the LCM of one of the two Find the LCM of one of the two variables.variables.

2.2. Multiply each individual equation by Multiply each individual equation by the necessary factor to cancel.the necessary factor to cancel.

3.3. Add the two equations if they have Add the two equations if they have opposite signs, in not then subtract.opposite signs, in not then subtract.

4.4. Solve for other variable.Solve for other variable.

5.5. Back substitute into other equation Back substitute into other equation to find other variableto find other variable

5x 3y 5

3x 2y 16

-3y and +2y could be turned into -6y and +6y

10x – 6y = 10

9x + 6y = -48ADD!!!19x = -38 x = -2NOW BACK SUBSTITUTE

A System Can Be . . . A System Can Be . . . If lines are parallel and answer If lines are parallel and answer

is null setis null set ThiThis is s is inconsistentinconsistent

If two lines cross at one pointIf two lines cross at one point This is This is ConsistentConsistent

When same line is used twiceWhen same line is used twice This is This is DependentDependent

4

2

-2

-4

4

2

-2

-4

4

2

-2

-4

Factoring

The 7 methods of Factoring: GCF Difference of Squares Sum & Difference of Cubes PST Reverse Foil Grouping 2x2 Grouping 3x1

Factoring Continued…..

Factoring with GCF: When factoring GCF is the first thing you look for if all

of the terms have a multiple in common you divide that out of each.

Factoring with Difference of Squares: WHEN THE SUM of two numbers multiplies their

difference -- (a + b)(a − b)

--- then the product is the difference of their squares: (a + b) (a − b) = a² − b²


Factoring with Sum & Difference of Cubes You can use the difference of squares rule to factor

binomials that can be written in the form a2 – b2. Sometimes the terms of a binomial have common factors. If so, the GCF should always be factored out first.

Formulas: a2 - b2 (a + b) (a - b) or (a - b) (a - b)

Factoring with PST (Perfect Square Trinomial) : If the square root of “a” and “c” can be found and if twice

their product is equal to middle term, then the trinomial can be factor out as Perfect Square Trinomial (PST).


Factoring with Reverse Foil: ax² + bx + c The difference between this trinomial and the one

discussed above, is there is a number other than 1 in front of the x squared. This means, that not only do you need to find factors of c, but also a.

Factoring by Grouping When factoring by grouping, you must first identify

patterns of common factors.

Need more help???

Inspiration file for more Factoring Help

Rational ExpressionsRational Expressions

Simplifying by factoring and cancelingSimplifying by factoring and canceling Ex: Ex: xx22 + 9x + 18 + 9x + 18

xx22 + 4x - 12 + 4x - 12

11stst – factor – factor (x + 6)(x + 3)(x + 6)(x + 3)

(x + 6)(x – 2)(x + 6)(x – 2)

22ndnd – cancel – cancel (x + 3)(x + 3)

(x – 2)(x – 2)

Rational Expressions (cont)Rational Expressions (cont)

Addition and Subtraction Addition and Subtraction 11stst – factor – factor 22ndnd – multiply by missing factor on top and – multiply by missing factor on top and

bottom of each equationbottom of each equation 33rdrd – simplify – simplify 44thth – look for more possible cancelation – look for more possible cancelation

factors factors Example problem- Example problem-

http://www.youtube.com/watch?v=omv7Di2o8-Y&feature=channel

Rational Expressions (cont) Rational Expressions (cont)

MultiplyingMultiplying 11stst – look for – look for

possible factoringpossible factoring 22ndnd – then cancel – then cancel

if possible from if possible from any top to any any top to any bottombottom

33rdrd – multiply – multiply acrossacross

Dividing1st – take

reciprocal of 2nd fraction 2nd – just multiply the ration

expressions

Quadratic Equations in One Quadratic Equations in One VariableVariable

First you must set equal to zero.

Then you factor.

Use Zero Product Property to finish the problem.


X2= 36

Take square root of both sides

X=6 (final answer)


Completing the SquareCompleting the Square EX: xEX: x22+6x-6=0+6x-6=0

Move the -6 to the other sideMove the -6 to the other side XX22+6x =6 (leave space to complete square)+6x =6 (leave space to complete square)

Take half of six and square it to complete squareTake half of six and square it to complete square xx22+6x+9=6+9+6x+9=6+9

The trinomial is a PST The trinomial is a PST (X+3)(X+3)22=15=15

Take the square root of both sidesTake the square root of both sides X+3=√15X+3=√15

Subtract three from both sidesSubtract three from both sides X=-3+-√15X=-3+-√15


• QUADRATIC FORMULA:

• The b, a, and c are coefficients

• Plug the numbers from you equation in for these letters

2-b± b -4acx =2a

Quadratic Equations in One VariableQuadratic Equations in One Variable

1. y = 2x2 – x - 6

2. f(x) = 2x2 – x + 6

3. y = -2x2– 9x + 6

4. f(x) = x2 – 6x + 9

1–4(2)(-6)=49 2 rational zerosopens up/vertex below x-axis/2 x-intercepts

1–4(2)(6)=-47 no real zerosopens up/vertex above x-axis/No x-intercepts

81–4(-2)(6)=129 2 irrational zerosopens down/vertex above x-axis/2 x-intercepts

36–4(1)(9)=0 one rational zeroopens up/vertex ON the x-axis/1 x-intercept

•What does the discriminant tell you?•The “zeros” of a function are the x-intercepts on it’s graph. Use the discriminant to predict how many x-intercepts each parabola will have and where the vertex is located.

Functions

What does f(x)= mean??? f (x)= means the same thing as y=

What is the domain and range??? Domain: The set of numbers x for which f(x) is

defined Range: The set of all the numbers f(x) for x in the

domain of “ f ”

Parabolas

Vertex: -b/2a x intercepts: given that y is zero for all x

intercepts plug 0 into where all of the y’s would be.

y intercepts: given that x is zero for all y intercepts plug 0 where all of the x’s would be.

Parabolas Continued….

Equation: y = x2 – 6x + 5 Vertex: (3, -4) y intercept: (0,5) x intercepts: (5,0) and

(1,0) Given these points your

graph would be:

Simplifying Expressions with Simplifying Expressions with ExponentsExponents

XX55 x X x X88=x=x13 13 (just add exponents together)(just add exponents together)

(3 x 4)2= (12)2= 144 (3 x 4)2= (12)2= 144 (PEMDAS)(PEMDAS)

(2(222))44 = 2 = 288 = 256 = 256 XX44/x/x33=x=x4-34-3 = x = x (just subtract) (just subtract)

(9/3)(9/3)22 = (3) = (3)22 = 9 = 9 (5x+5y+7m)(5x+5y+7m)00 = 0 = 0 XX-2-2 = 1/X = 1/X22

Simplifying Expressions with Simplifying Expressions with RadicalsRadicals

√√5 x √7 = √355 x √7 = √35 √√1/√2 = √1/√2 x √2/√2 = √2/2 1/√2 = √1/√2 x √2/√2 = √2/2 (rationalizing (rationalizing

the denominator)the denominator)

√√xx22 = x = x (the root and power cancel each other out) (the root and power cancel each other out)

22√4 = 4 √4 = 4

Word Problems!!!!

A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If there are three times as many nickels as quarters, and one-half as many dimes

as nickels, how many coins of each kind are there? number of quarters: q

number of nickels: 3q number of dimes: (½)(3q) = (3/2)q

There is a total of 33 coins, so: q + 3q + (3/2)q = 33

4q + (3/2)q = 33 8q + 3q = 66 11q = 66

q = 6

Word Problems!!!!

A triangle has a perimeter of 50. If 2 of its sides are equal and the third side is 5 more than the equal sides, what is the length of the third side? Step 1: Assign variables: Let x = length of the equal side

Sketch the figure

x x

x +5

Continued….

Last problem continued…

Plug in the values from the question and from the sketch: 50 = x + x + x+ 5

Combine like terms: 50 = 3x + 5 Isolate variable x:

3x = 50 – 5 3x = 45x =15

The length of third side = 15 + 5 =20

Answer: The length of third side is 20

Word Problems!!!!

Polar bears are extremely good swimmers and can travel as long as 10 hours without resting. If a polar bear is swimming with an average speed of 2.6 m/s, how far will it have traveled after 10.0 hours? speed, v = 2.6 m/s time, t = 10.0 h = 10.0 h × 3600 s/h = 3.6 × 104 s Unknown: distance, d = ?

d =2 .6 m / s × (3.6 × 104 s) d = 9.4 × 104 m = 94 km

Word Problems!!!!

Florence Griffith-Joyner set the women’s world record for running 200.0 m in 1988. At the 1988 Summer Olympics in Seoul, South Korea, she completed the distance in 21.34 s. What was Florence Griffith-Joyner’s average speed? Substitute distance and time values into the speed

equation, and solve. v = d / t = 200.0 m / 21.34 s

v = 9.372 m/s

Line of Best Fit or Regression Line A line of best fit (or "trend"

line) is a straight line that best represents the data on a scatter plot. This line may pass through some of the points, none of the points, or all of the points.

Given this set of data:

SandwichTotal Fat (g)

Total Calori

es

Hamburger 9 260

Cheeseburger 13 320

Quarter Pounder 21 420

Quarter Pounder with Cheese

30 530

Big Mac 31 560

Arch Sandwich Special 31 550

Arch Special with Bacon

34 590

Crispy Chicken 25 500

Fish Fillet 28 560

Grilled Chicken 20 440

Grilled Chicken Light 5 300For graphing on calculator view this

Line of Best Fit or Regression Line Continued…. Your best fit line would look like this:

You use the line of

best fit when your

data is scattered.

650

600

550

500

450

400

350

300

250

200

150

100

50

y

THE END!!!!!

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Reviewing Can’t Hurt By: James Christian and Jack Gryniewicz