Rational Expressions and Equations

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Rational Expressions and Equations

• Chapter 7

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Chapter Sections
7.1 Simplifying Rational Expressions 7.2
Multiplying and Dividing Rational Expressions 7.3
Adding and Subtracting Rational Expressions
with a Common Denominator 7.4 Finding the Least
Common Denominator and Forming Equivalent
Rational Expressions 7.5 Adding and Subtracting
Rational Expressions with Unlike Denominators 7.6
Complex Rational Expressions 7.7 Rational
Equations 7.8 Models Involving Rational
Equations
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Simplifying Rational Expressions

• Section 7.1

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Rational Expressions
To evaluate a rational expression, replace the
variable with its assigned numerical value and
perform the arithmetic.
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Evaluating Rational Expressions
Example Evaluate for
x 1 and y 3.
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Undefined Values
Because a rational expression is undefined for
those values of the variable(s) that make the
denominator zero, we find the values for which a
rational expression is undefined by setting the
denominator equal to zero and solving for the
variable.
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Simplifying Rational Expressions
Simplifying Rational Expressions If p, q, and r
are polynomials, then
if q ? 0 and r ? 0.
Factor the numerator.
Divide out the common factor.
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Simplifying Rational Expressions
Example Simplify
Factor x from the numerator.
Factor the numerator and denominator.
Divide out the common factor.
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Multiplying and Dividing Rational Expressions

• Section 7.2

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Multiplying Rational Expressions
Steps to Multiply Rational Expressions Step 1
Factor the polynomials in each numerator and
denominator. Step 2 Use the fact that if are
two rational expressions, then to
multiply the rational expressions. Step 3
Divide out common factors in the numerator and
denominator. Leave your answer in factored form.
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Multiplying Rational Expressions
Example Multiply
Multiply.
1
5
1
Divide out common factors.
9
4
x2
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Multiplying Rational Expressions
Example Multiply
Factor each numerator and denominator.
Factor again whenever possible.
Divide out common factors.
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Dividing Rational Expressions
Steps to Divide Rational Expressions Step 1
Multiply the dividend by the reciprocal of the
divisor. Step 2 Factor each polynomial in the
numerator and denominator. Step 3
Multiply. Step 4 Divide out common factors in
the numerator and denominator. Leave the
remaining factors in factored form.
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Dividing Rational Expressions
Example Divide
Rewrite the division.
Multiply the numerator by the reciprocal of the
denominator.
2
Divide out common factors.
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Dividing Rational Expressions
Example Divide
Invert the second fraction and multiply.
Factor the numerator and denominator.
Divide out common factors.
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Adding and Subtracting Rational Expressions with
a Common Denominator

• Section 7.3

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Adding Rational Expressions
Adding Rational Expressions with a
Common Denominator Step 1 Use the fact that if
are two rational
expressions, then
to add the expressions. Step 2 Simplify
the sum by writing the rational expression in
lowest terms. This step will not always be
necessary.
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Adding Rational Expressions
Example Find the sum
and simplify, if possible.
Add the numerators.
Factor the numerator and denominator.
Divide out like factors.
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Subtracting Rational Expressions
Subtracting Rational Expressions with a
Common Denominator Step 1 Use the fact that if
are two rational
expressions, then
to subtract the expressions. Step 2
Simplify the difference by writing the rational
expression in lowest terms. This step will not
always be necessary.
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Subtracting Rational Expressions
Example Find the difference
and simplify, if possible.
Subtract the numerators.
Simplify.
Factor the numerator and denominator.
Divide out like factors.
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Finding the Least Common Denominator and Forming
Equivalent Rational Expressions

• Section 7.4

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Finding the LCD
If the denominators of a rational expression are
not the same, the rational expression must be
written using a least common denominator. A
least common denominator (LCD) is the smallest
polynomial that is a multiple of each denominator
in the expression to be added or subtracted.
After factoring the denominators, we can see that
the LCD is (2)(3)(3 w) 6(3 w).
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Finding the LCD
Finding the Least Common Denominator Step 1
Factor each denominator completely. When
factoring, write the factored form using powers.
For example, write x2 4x 4 as (x 2)2. Step
2 If the factors are common except for their
power, then list the factor with the highest
power. That is, list each factor the greatest
number of times that it appears. Then list the
uncommon factors. Step 3 The LCD is the product
of the factors written in Step 2.
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Finding the LCD
Example Find the LCD of the rational
expressions .
Factor each denominator.
Use the factor that is repeated the greatest
number of times.
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Finding Equivalent Expressions
Steps to Form Equivalent Rational
Expressions Step 1 Write each denominator in
factored form. Step 2 Determine the missing
factor(s). That is, what factor(s) does the new
denominator have that is missing from the
original denominator? Step 3 Multiply the
original rational expression by Step 4 Find
the product. Leave the denominator in factored
form.
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Finding Equivalent Expressions
Example Write as an equivalent fraction
with a denominator of 48.
We want to change the denominator of 8 into a
denominator of 48.
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Finding Equivalent Expressions
Example Write the rational expression
with a denominator of x2y2z.
We want to change the denominator of xyz to a
denominator of x2y2z.
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Writing Equivalent Expressions
Example Find the LCD of the rational
expressions Rewrite each expression.
x2 x x(x 1)
x2 x 2 (x 2)(x 1)
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Adding and Subtracting Rational Expressions with
Unlike Denominators

• Section 7.5

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Adding with Unlike Denominators
Adding or Subtracting Rational Expressions
with Unlike Denominators Step 1 Find the least
common denominator. Step 2 Rewrite each
rational expression with the common denominator.
You will need to multiply out the numerator, but
leave the denominator in factored form. Step 3
Add or subtract the rational expressionsfound in
Step 2. Step 4 Simplify the result, if possible.
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Adding with Unlike Denominators
Example Find the sum
The LCD is xy.
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Adding with Unlike Denominators
Example Find the sum
The LCD is 112x2y4.
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Subtracting with Unlike Denominators
Example Find the difference
The LCD is 2(x 3).
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Subtracting with Unlike Denominators
Example Find the difference
The LCD is (d 6)2(d 6).
Continued.
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Subtracting with Unlike Denominators
Example continued
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Complex Rational Expressions

• Section 7.6

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Simplifying Rational Expressions
A complex rational expression is a fraction in
which the numerator and/or the denominator
contains the sum or difference of rational
expressions.
1. Simplifying the numerator and the denominator
separately, or 2. Using the least common
denominator.
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Simplifying Rational Expressions
Simplifying a Complex Rational Expression by
Simplifying the Numerator and Denominator
Separately (Method I) Step 1 Write the
numerator of the complex rational expression as a
single rational expression. Step 2 Write the
denominator of the complex rational expression as
a single rational expression. Step 3 Rewrite
the complex rational expression using the
rational expressions determined in Steps 1 and
2. Step 4 Simplify the rational expression
using the techniques for dividing rational
expressions from Section 5.2.
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Simplifying Rational Expressions
Example Simplify
Write the denominator as a single expression.
Divide.
Continued.
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Simplifying Rational Expressions
Example continued
Simplify.
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Simplifying Rational Expressions
Simplifying a Complex Rational Expression
by Using the Least Common Denominator (Method
II) Step 1 Find the least common denominator
among all the denominators in the complex
rational expression. Step 2 Multiply both the
numerator and denominator of the complex rational
expression by the least common denominator found
in Step 1. Step 3 Simplify the rational
expression, if possible.
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Using the LCD to Simplify
Example Simplify
The LCD of all the denominators is 8w.
Multiply each term by 8w.
Simplify.
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Rational Equations

• Section 7.7

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Solving Equations
A rational equation is an equation that contains
a rational expression.
Multiply each term by the LCD 6x.
Simplify.
Subtract 2 from both sides.
Divide both sides by 5.
?
Check
The solution set is 2.
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Solving Equations
Solving a Rational Equation Step 1 Determine
the values(s) of the variable that result in any
undefined rational expression in the rational
equation. Step 2 Determine the least common
denominator (LCD) of all the denominators. Step
3 Multiply both sides of the equation by the
LCD, and simplify the expression on each side of
the equation. Step 4 Solve the resulting
equation. Step 5 Verify your solution using the
original equation.
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Solving Equations
Example Solve
Multiply each term by the LCD (y 5)(3y 2).
Simplify.
Distribute to remove parentheses.
Add 6 to both sides.
Continued.
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Solving Equations
Example continued
Subtract 3y from both sides.
Divide both sides by 6 and simplify.
Check
?
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Solving Equations with No Solutions
Example Solve
a ? 6
Multiply each term by (a 6).
Simplify.
Distribute.
Add 5a to both sides.
Add 10 to both sides.
Continued.
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Solving Equations with No Solutions
Example continued
Divide both sides by 7.
Extraneous solution
a 6 is not in the domain of the variable a, so
there is no solution to the equation and the
solution set is or ?.
Check
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Solving for a Variable
Example In the following finance formula, solve
for r
Multiply each side by 1 r.
Distribute and simplify.
Subtract P from both sides.
Divide both sides by P.
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Models Involving Rational Equations

• Section 7.8

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Ratio and Proportion
A ratio is the quotient of two numbers or two
quantities. The ratio of two numbers a and b can
be written as
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Solving a Proportion
Example Solve the proportion
The LCD is 20.
Divide out common factors.
Divide both sides by 2.
?
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Cross-Multiplication Property
Another method used to solve a proportion is the
method of cross multiplication.
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Solving a Proportion
Example Solve the proportion
Find the cross products.
Divide each side by 4.
?
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Similar Figures
Two figures are similar if their corresponding
angle measures are equal and their corresponding
sides are proportional.
Example Triangles A and B are similar. Find the
length of side y.
The ratio of the base of Triangle A to the short
side of A equals the ratio of the base of
Triangle B to the short side of B.
Continued.
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Similar Figures
Example continued.
Multiply both sides of the equation by the LCD,
28.
Simplify.
The length of the base of Triangle B is 34.
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Solving Work Problems

• Example
• It takes John 4 hours to paint a fence. When his
  friend, Tom, helps him and the two work together,
  the fence can be painted in 3 hours. How long
  would it take Tom to paint the fence by himself?

Step 1 Identify
Continued.
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Solving Work Problems
Example continued
Step 2 Name
Step 3 Translate
Continued.
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Solving Work Problems
Example continued
Step 4 Solve
Multiply each term by the LCD, 12x.
Simplify.
Subtract 3x from each side.
Working alone, it would take Tom 12 hours to
paint the fence.
Step 5 Check
?
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Solving Motion Problems
Example Nancy drove her car to Cleveland while
Patty drove her car to Columbus. Nancy drove 360
kilometers while Patty drove 280 kilometers.
Nancy drove 20 kilometers per hour faster than
Patty on her trip. What was the average speed in
kilometers per hour for each driver?
Step 1 Identify
Distance problems can be solved using the
formula distance rate time (d rt).
Step 2 Name
Let r the rate of Pattys car.
Let r 20 the rate of Nancys car.
The time, t, for each driver was the same.
Continued.
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Solving Motion Problems
Example continued
Step 3 Translate
Step 4 Solve
Since the time for each driver was the same, we
can set the times equal to each other.
Continued.
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Solving Motion Problems
Example continued
Multiply both sides by r(r 20).
Simplify.
Distribute.
Subtract 280r from each side.
Divide each side by 80.
Pattys rate was 70 kilometers per hour. Nancys
rate was 90 kilometers per hour.
Continued.
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Solving Motion Problems
Example continued
Step 5 Check
?
Pattys speed was 70 kph. Nancys speed was 90
kph.
Nancys speed was 20 kph faster than Pattys.
?