Lesson 8-1: Multiplying and Dividing Rational Expressions

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Lesson 8-1 Multiplying and Dividing Rational
Expressions
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Rational Expression

• Definition a ratio of two polynomial expressions

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To Simplify A Rational Expression

• 1. Make sure both the numerator and denominator
  are factored completely!!!
• 2. Look for common factors and cancel
• Remember factors are things that are being
  multiplied you can NEVER cancel things that are
  being added or subtracted!!!
• 3. Find out what conditions make the expression
  undefined and state them.

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Examples Simplify and state the values for x
that result in the expression being undefined
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2.
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Examples Cont Simplify
3.
5.
4.
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Operations with Rational Expressions
To Multiply Rational Expressions Factor and
cancel where possible. Then multiply numerators
and denominators
Define x-values for which the expression is
undefined
To Divide Rational Expressions Rewrite the
problem as a multiplication problem with the
first expression times the reciprocal of the
second expression. Then factor and cancel where
possible. Multiply numerators and denominators
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Examples Simplify
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Polynomials in Numerator and Denominator

• Rules are the same as before
• 1. Make sure everything is factored completely
• 2. Cancel common factors
• 3. Simplify and define x values for which the
  expression is undefined.

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Examples Simplify and define x values for which
it is undefined
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Examples
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Simplifying complex fractions

• A complex fraction is a rational expression whose
  numerator and/or denominator contains a rational
  expression

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To simplify complex fractions

• Same rules as before
• Rewrite as multiplication with the reciprocal
• Factor and cancel what you can
• Simplify everything
• Multiply to finish

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Examples
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Lesson 8-2 Adding and Subtracting Rational
Expressions
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Adding and Subtracting Rational Expressions

• Finding Least Common Multiples and Least Common
  Denominators!
• Use prime factorization
• Example Find the LCM of 6 and 4
• 6 23
• 4 22
• LCM 223 12

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Find the LCM

• 1. 18r2s5, 24r3st2, and 15s3t
• 2. 15a2bc3, 16b5c2, 20a3c6
• 3. a2 6a 9 and a2 a -12
• 4. 2k3 5k2 12k and k3 8k2 16k

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Add and Subtract Rational Expressions

• Same as fractions
• To add two fractions we find the LCD, the same
  things is going to happen with rational
  expressions

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Examples Simplify
5.
6.
7.
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9.
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Lesson 8-3 Graphing Rational Functions
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Definitions

• Continuity graph may not be able to be traced
  without picking up pencil
• Asymptote a like that the graph of the function
  approaches, but never touches (this line is
  graphed as a dotted line)
• Point discontinuity a hole in the graph

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Vertical Asymptote
How to find a Vertical Asymptote x the
value that makes the rational expression
undefined Set the denominator of the
rational expression equal to zero and solve.
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Point Discontinuity
How to find point discontinuity Factor
completely Set any factor that cancels
equal to zero and solve. Those are the x values
that are points of discontinuity
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Graphing Rational Functions

• f(x)

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Graphing Rational Functions

• f(x)

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Graphing Rational Functions

• f(x)

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Graphing Rational Functions

• f(x)

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Graphing Rational Functions

• f(x)

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Graphing Rational Functions

• f(x)

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Lesson 8-4 Direct, Joint, and Inverse Variation
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Direct Variation

• y varies directly as x if there is a nonzero
  constant, k, such that y kx
• k is called the constant of variation
• Plug in the two values you have and solve for the
  missing variable
• Plug in that variable and the other given value
  to solve for the requested answer

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Example

• If y varies directly as x and y 12 when
• x -3, find y when x 16.

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Joint Variation

• y varies jointly as x and z if there is a nonzero
  constant, k, such that y kxz
• Follow the same directions as before

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Example

• Suppose y varies jointly as x and z. Find y when
  x 8 and z 3, if y 16 when z 2 and x 5.

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Inverse Variation

• y varies inversely as x if there is a nonzero
  constant, k, such that xy k or y
• 
  

k x
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Example

• If y varies inversely as x and y 18 when x
  -3, find y when x -11

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Lesson 8-6 Solving Rational Equations and
Inequalities
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Lets review some old skills

• How do you find the LCM of two monomials
• 8x2y3 and 18x5
• Why do we find LCMs with rational expressions?

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Old Skills Cont

• What is it called when two fractions are equal to
  each other?
• What process do we use to solve a problem like
  this?

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To solve a rational equation

• 1. Make sure the problem is written as a
  proportion
• 2. Cross Multiply
• 3. Solve for x
• 4. Check our answer

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Examples

• Solve

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Lets put those old skills to new use
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Examples

• Solve

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Lesson 8-6 Day 2

• Rational Inequalities

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Solving Rational Inequalities

• Step 1 State any excluded values (where the
  denominator of any fraction could equal zero)
• Step 2 Solve the related equation
• Step 3 Divide a numberline into intervals using
  answers from steps 1 and 2 to create intervals
• Step 4 Test a value in each interval to
  determine which values satisfy the inequality

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Examples

• Solve

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Examples

• Solve

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Examples

• Solve

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Examples

• Solve