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

Solving 1st Power Equations in One Variable Example problem: 5(4+n) = 2(9+2n) 5(4+n) = 2(9=2n) Distribute the 5 and the 2 20+5n = 18+4n Subtract 4n from both sides 20+n = 18 Subtract the 20 from both sides to get n alone n = -2

Solving 1st Power Equations in One Variables (Cont’) Fractional Coefficients- Example: 1st: get rid of denominator by multiplying by 2 2nd: divide by 1 on both sides YOUR ANSWER IS 1

Solving 1st Power Equations in One Variables (Cont’) Variables in the Denominator-Example: Multiply by the LCD on both sides 9(3-x) Distribute Solve for x

Solving 1st Power Equations in One Variables (Cont’) Special Cases Variables cancels and you get the same number on both sides of the equal sign ALL REALS 2. Variables cancel but the numbers on either side of the equal sign are not the same

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 Inverse Property of Addition 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

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 Power of a Product 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 Power of a Quotient 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 The Zero Product 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 Quotient 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. = ----

Properties Root of a Power Property Power of a Root 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.

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 ● ____ ANSWERS 1. R 2. V 3. D 4. J 5. T 6. S 7. E 8. X 9. W 10. N 11. L 12. I

Inequality Rules When the sign has the equal to option the graph will have a closed circle and when the sign is only greater than or less than the graph has an open circle. Disjunction is an “or” statement when both statements can be true. Satisfies one or both statements. A conjunction is when both statements are true and the statement has an “and” in it.

Special Cases of Inequalities Watch for no solution- conjunction does not combine, but rather looks like a disjunction when graphed. Watch for when every number works- disjunction looks like a conjunction when graphed. One arrow- disjunction when one arrow proves the other to be right as well.

Solving Inequalities in One Variable Don’t forget the Multiplication Property of Inequality! If you multiply or divide by a negative, the sign must be reversed. -5x 10 Solution Set: {x: x > -2} x -2 SIGN REVERSAL!!! -2

Try Solving These Inequalities on Your Own Disjunction- 5x > 25 or 3x < 9 Conjunction- -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)

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.

How many points do these lines have in common? Linear Systems How many points do these lines have in common? Think about: 2x + y = 5 or y = - 2x + 5

How to Find the Common Points in Linear Systems Methods- Substitution- goal is to get one variable equal to an equation and substitute that expression into the other equation for that variable.

How to Find the Common Points in Linear Systems Method 2- Estimate the SOLUTION of a SYSTEM on a graph. Where do they intersect? THE SOLUTION IS : (2,1)

How to Find the Common Point of Linear Systems Method 3- Elimination or Addition/Subtraction -Goal is to combine equations in order to cancel one variable

ADD!!! 19x = -38 x = -2 Steps to solving: Find the LCM of one of the two variables. Multiply each individual equation by the necessary factor to cancel. Add the two equations if they have opposite signs, in not then subtract. Solve for other variable. Back substitute into other equation to find other variable -3y and +2y could be turned into -6y and +6y 10x – 6y = 10 9x + 6y = -48 ADD!!! 19x = -38 x = -2 NOW BACK SUBSTITUTE

A System Can Be . . . If lines are parallel and answer is null set This is inconsistent If two lines cross at one point This is Consistent When same line is used twice This is Dependent

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 Continued….. 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 Continued….. 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 Expressions Simplifying by factoring and canceling Ex: x2 + 9x + 18 x2 + 4x - 12 1st – factor (x + 6)(x + 3) (x + 6)(x – 2) 2nd – cancel (x + 3) (x – 2)

Rational Expressions (cont) Addition and Subtraction 1st – factor 2nd – multiply by missing factor on top and bottom of each equation 3rd – simplify 4th – look for more possible cancelation factors Example problem- http://www.youtube.com/watch?v=omv7Di2o8-Y&feature=channel

Rational Expressions (cont) Dividing 1st – take reciprocal of 2nd fraction 2nd – just multiply the ration expressions Multiplying 1st – look for possible factoring 2nd – then cancel if possible from any top to any bottom 3rd – multiply across

Quadratic Equations in One Variable First you must set equal to zero. Then you factor. Use Zero Product Property to finish the problem.

Quadratic Equations in One Variable X2= 36 Take square root of both sides X=6 (final answer)

Quadratic Equations in One Variable Completing the Square EX: x2+6x-6=0 Move the -6 to the other side X2+6x =6 (leave space to complete square) Take half of six and square it to complete square x2+6x+9=6+9 The trinomial is a PST (X+3)2=15 Take the square root of both sides X+3=√15 Subtract three from both sides X=-3+-√15

Quadratic Equations in One Variable QUADRATIC FORMULA: The b, a, and c are coefficients Plug the numbers from you equation in for these letters

Quadratic Equations in One Variable 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. 1. y = 2x2 – x - 6 1–4(2)(-6)=49  2 rational zeros opens up/vertex below x-axis/2 x-intercepts 2. f(x) = 2x2 – x + 6 1–4(2)(6)=-47  no real zeros opens up/vertex above x-axis/No x-intercepts 3. y = -2x2– 9x + 6 81–4(-2)(6)=129 2 irrational zeros opens down/vertex above x-axis/2 x-intercepts 4. f(x) = x2 – 6x + 9 36–4(1)(9)=0  one rational zero opens up/vertex ON the x-axis/1 x-intercept

Functions What does f(x)= mean??? What is the domain and range??? 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 Exponents X5 x X8=x13 (just add exponents together) (3 x 4)2= (12)2= 144 (PEMDAS) (22)4 = 28 = 256 X4/x3=x4-3 = x (just subtract) (9/3)2 = (3)2 = 9 (5x+5y+7m)0 = 0 X-2 = 1/X2

Simplifying Expressions with Radicals √1/√2 = √1/√2 x √2/√2 = √2/2 (rationalizing the denominator) √x2 = x (the root and power cancel each other out) 2√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!!!!      x x Continued…. Step 1: Assign variables: 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 = 45 x =15 The length of third side = 15 + 5 =20 Answer: The length of third side is 20

Word Problems!!!!      d =2 .6 m / s × (3.6 × 104 s) 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: Sandwich Total Fat (g) Total Calories Hamburger 9 260 Cheeseburger 13 320 Quarter Pounder 21 420 Quarter Pounder with Cheese 30 530 Big Mac 31 560 Arch Sandwich Special 550 Arch Special with Bacon 34 590 Crispy Chicken 25 500 Fish Fillet 28 Grilled Chicken 20 440 Grilled Chicken Light 5 300 For 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.

THE END!!!!!