Rational Expressions and Equations

1
Rational Expressions and Equations

• Chapter 7

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Chapter Sections

• 7.1 Simplifying Rational Expressions
• 7.2 Multiplication and Division of Rational
  Expressions
• 7.3 Addition and Subtraction of Rational
  Expressions with a Common Denominator
• 7.4 Addition and Subtraction of Rational
  Expressions
• 7.5 Complex Fractions
• 7.6 Solving Rational Equations
• 7.7 Rational Equations Applications Problem
  Solving
• 7.8 Variation

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7.1

• Simplifying Rational Expressions

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Rational Expressions

• A rational expression is an expression of the
  form p/q where p and q are polynomials and q ? 0.

Examples
Whenever a rational expression contains a
variable in the denominator, assume that the
values that make the denominator 0 are excluded.
5
Signs of a Fraction

• Three signs are associated with any fraction
  the sign of the numerator, the sign of the
  denominator, and the sign of the fraction.

Changing any two of the three signs of a
fraction does not change the value of the fraction
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Simplifying

• A rational expression is simplified or reduced
  to lowest terms when the numerator and
  denominator have no common factors other than 1.

Examples
7
Simplifying Rational Expressions

• Factor both the numerator and denominator as
  completely as possible.
• Divide out any factors common to both the
  numerator and denominator.

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Factoring a Negative 1

• Remember that when 1 is factored from a
  polynomial, the sign of each term in the
  polynomial changes.

Example 2x 5 1(2x 5) (2x 5)
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7.2

• Multiplication and Division of Rational
  Expressions

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Multiplying Fractions
1
7
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Multiplying Rational Expressions

• Factor all numerators and denominators
  completely.
• Divide out common factors.
• Multiply numerators together and multiply
  denominators together.

12
Dividing Two Fractions
1
1
13
Dividing Rational Expressions

• Invert the divisor (the second fraction) and
  multiply

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7.3

• Addition and Subtraction of Rational Expressions
  with a Common Denominator

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Adding/Subtracting Fractions
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Common Denominators

• Add or subtract the numerators.
• Place the sum or difference of the numerators
  found in step 1 over the common denominator.
• Simplify the fraction if possible.

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Common Denominators

• Example

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Common Denominators

• Example

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Least Common Denominator

• Factor each denominator completely. Any factors
  used more than once should be expressed as
  powers.
• List all different factors that appear in any of
  the denominators. When the same factor appears
  in more than one denominator, write that factor
  with the highest power that appears.
• The least common denominator (LCD) is the product
  of all the factors listed in step 2.

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Least Common Denominator

• Find the LCD

The LCD is (2x 5)(x 5).
The LCD is x(x 1).
The LCD is 36w5z4.
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7.4

• Addition and Subtraction of Rational Expressions

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Unlike Denominators

• Determine the LCD.
• Rewrite each fraction as an equivalent fraction
  with the LCD.
• Add or subtract the numerators while maintaining
  the LCD.
• When possible, factor the remaining numerator and
  simplify the fraction.

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Unlike Denominators

• Example

The LCD is w(w2).
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Unlike Denominators

• Example

The LCD is 12x(x 1).
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Unlike Denominators

• Example

The LCD is (x 5)(x2).
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7.5

• Complex Fractions

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Simplifying Complex Fractions

• A complex fraction is one that has a fraction in
  its numerator or its denominator or in both the
  numerator and denominator.

Example
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Simplify by Combining Like Terms
Method 1

• Add or subtract the fraction in both the
  numerator and denominator of the complex fraction
  to obtain single fractions in both the numerator
  and the denominator.
• Invert the denominator of the complex fraction
  and multiply the numerator by it.
• Simplify further if possible.

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Simplify by Combining Like Terms

• Simplify

30
Simplify by Multiplying
Method 2

• Find the LCD of all the denominators appearing in
  the complex fraction.
• Multiply both the numerator and the denominator
  of the complex fraction by the LCD.
• Simplify further if possible.

31
Simplify by Multiplying

• Simplify

32
7.6

• Solving Rational Equations

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Complex Fractions

• A rational equation is one that contains one or
  more rational (fractional) expressions.

Example
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Solving Rational Equations

• Find the LCD of all fractions in the equation.
• Multiply both sides of the equation by the LCD.
  (Every term will be multiplied by the LCD.)
• Remove any parentheses and combine like terms on
  each side of the equation.
• Solve the equation.
• Check the solution in the original equation.

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Integer Denominators
Solve the equation
The LCD is 30.
?
CHECK
36
Variable Denominators

• Whenever there is a variable in the denominator,
  it is necessary to check your answer in the
  original equation. If the answer obtained makes
  the denominator zero, that value is NOT a
  solution to the equation.

The LCD is 2z.
CHECK
?
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Variable Denominators
The LCD is 2(x-3).
CHECK
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7.7

• Rational Equations Applications Problem
  Solving

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Geometry Applications
Yield signs are triangles. The area of the sign
is about 558 square inches. The height of the
sign is about 5 inches less than its base.
Determine the length of the base of the yield
sign.
Solve the equation.
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Geometry Applications
The base is 36 inches.
?
CHECK
41
Work Problems

• Problems in which two or more people or machines
  work together to complete a certain task are
  referred to as work problems.

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Work Problems
Example

• At the NCNB Savings Bank it takes a computer 4
  hours to process and print payroll checks. When
  a second computer is used and the two computers
  work together, the checks can be printed in 3
  hours. How long would it take the second
  computer by itself to process and print the
  payroll checks?

Continued.
43
Work Problems
Example continued
A table helps to keep information organized.
Continued.
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Work Problems
Example continued
Solve the equation.
Continued.
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Work Problems
Example continued
The LCD is 4x.
It would take the second computer 12 hours by
itself.
?
CHECK
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7.8

• Variation

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Variation

• A variation is an equation that relates one
  variable to one or more other variables using the
  operations of multiplication or division. There
  are two different types of variation

direct variation and inverse variation
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Direct Variation

• The formula used to calculate distance is
• distance rate ? time
• If the rate is a constant 50 miles per hour, the
  formula can be written as
• d 50t
• The distance, d, varies directly as the time, t,
  so
• the distance is directly proportional to the
  time, t.
• The 50 is the constant of proportionality.

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Direct Variation

• If a variable y varies directly as a variable x,
  then
• y kx
• where k is the constant of proportionality (or
  the variation constant).

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Direct Variation

• Example
• s varies directly as the square of m.
• If s 100 when m 5, find s when m 12.

s km2 100 k(52) 100 25k 4 k
First, determine what k represents.
Then determine what s is when m 12.
s km2 s 4(122) s 576
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Indirect Variation

• If a variable y varies inversely as a variable
  x, then
• where k is the constant of proportionality.

Example R varies inversely as W. Find R
when W 160 and k 240.