Chapter+12

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= 12-1 =
 * Graph inverse variations
 * Solve problems involving inverse variation

y varies inversely as x if there is some nonzero constant k such that xy=k

If k value is positive you will use quadrants one and three

Product rule! For Inverse Variation- if (x1, y1) and (x2, y2) are solutions then (x1,y1) = (x2,y2) because both = k.

//You can use this equation to solve for missing values// //.// **Inverse Variation** In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.

For instance, a biker traveling at 8 mph can cover 8 miles in 1 hour. If the biker's //speed decreases// to 4 mph, it will take the biker 2 hours (//an increase of one hour)//, to cover the same distance.

Inverse variation: when one variable //increases//,the other variable //decreases.//

Notice the shape of the graph of inverse variation.If the value of //x// is increased,then //y// decreases. If //x// decreases, the //y// value increases. We say that //y// varies inversely as the value of //x//.

**Direct Variation (not in this section)**
When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct variation. It is said that one variable "varies directly" as the other.

The constant ratio is called the constant of variation.

The formula for direct variation is //y = kx,// where //k// is the constant of variation."//y// varies directly as //x//"

In a direct variation, the two variables change in the same sense. If one increases, so does the other.

= 12-2 =
 * Identify values excluded from the domain of a rational expression
 * Simplify rational expressions

Rational Expression- an algebraic fraction whose numerator and denominator are polynomials

Excluded values- values that would make the denominator 0 (possibility of none- if there are no variables on the bottom, there is no excluded values)

** __Steps:__ **
Ignore the numerator Set the denominator equal to zero Find solution set (contains excluded values)
 * // Easy //**

Prime factorization Simplify
 * // Harder //**

1. Factor both the top and bottom of the algebraic fraction. ................. A. Put the terms in descending order (if there are three or more terms). ................. B. If all numbers are divisible by one number, simplify ................. B. Factor out the GCF (if there is one). ................. C. Evaluate which type of polynomial it is. ....................... 1. If there are 2 terms use difference of squares (if possible) ....................... 2. If there are 3 terms check for: ............................ - perfect square trinomials ............................ - target numbers ....................... 3. If there are 4 or more terms use grouping (if possible). 2. Simplify like terms. 3. Solve for the variable in the denominator (before it was simplified) to find out what the variable cannot be. The denominator cannot be zero.
 * // (More Complex Problems) __If there is adding/subtracting signs on top you factor both top and bottom!__ //**

__//**Factoring Hints:**//__

**Difference of squares** OR
 * squared variable and number that can be squared
 * two squared variables
 * MUST have a subtraction sign


 * Polynomial With GCF for each term **
 * same variables
 * common factors
 * two variables (one squared and one not squared)


 * Trinomials **
 * three terms
 * two operations
 * Ax^2 + Bx + C (multiply ac - multiply to the product of ac and add to b - split the b -factor by grouping)
 * x^2 + Bx + C (multiply to find c and add to find b)

= 12-3 =
 * Multiply rational expressions
 * Use dimensional analysis with multiplication

1. Factor both the top and bottom of both algebraic fractions. ....... A. Put the terms in descending order (if there are three or more terms). ....... B. Factor out the GCF (if there is one). ....... C. Evaluate which type of polynomial it is. ............... 1. If there are 2 terms use difference of squares (if possible) ............... 2. If there are 3 terms check for: ............................. -perfect square trinomials ............................. -target numbers ............... 3. If there are 4 or more terms use grouping (if possible). 2. Simplify by canceling like terms. 3. Multiply your remaining numerators together. 4. Multiply your remaining denominators together. 5. Simplify .

//** To make it easier simplify first then multiply numerator and denominator **//
//** Dementional Analysis **//


 * 1) Set up units in correct order
 * 2) Plug in numbers
 * 3) Simplify
 * 4) Multiply numerator
 * 5) Multiply denominator
 * 6) Make sure denominator is 1



= 12-4 =
 * Divide rational expressions
 * Use dimensional analysis with division

//** When it's fractions you're dividing by, flip the second number upside down and multiply **//





= 12-6 =


 * Add rational expressions with like denominators
 * Subtract rational expressions with like denominators

= 12-7 =
 * Add rational expressions with unlike denominators
 * Subtract rational expressions with unlike denominators

//__ Steps ____ : __//
 * 1) // wrote the problem //
 * 2) // factored the denominator (to find the common denominator) //
 * 3) // multiplied the fractions by 1 (to make common denominators) //

= 12-8 =
 * Add rational expressions with unlike denominators
 * Subtract rational expressions with unlike denominator

Mixed Fraction
 * 1) Put the number over 1 (Change into a mixed fraction)
 * 2) Get common denominator by multipying by the other denominator (pull apart fraction and divide both)
 * 3) Add numerators
 * 4) Combine like terms and put in descending order

(Change into a mixed fraction) (Pull apart fraction and divide both) (When it's fractions your dividing by turn the second number upside down and multiply) (Simplify!)

= 12-9 =
 * Solve rational equations
 * Eliminate extraneous solutions

As soon as you have a variable in the denominator your problem is equal to zero

Extranious Solution: When you plug a number in and it makes the number equal to zero