Warm Up

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Preview
Warm Up
California Standards
Lesson Presentation
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• Warm Up
• Add. Simplify your answer.
• 1. 2.
• 3. 4.

Subtract. Simplify your answer.
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6.
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The rules for adding rational expressions are the
same as the rules for adding fractions. If the
denominators are the same, you add the numerators
and keep the common denominator.
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Additional Example 1A Adding Rational
Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator. Divide out
common factors.
Simplify.
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Additional Example 1B Adding Rational
Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.
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Additional Example 1C Adding Rational
Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.
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Check It Out! Example 1a
Add. Simplify your answer.
Combine like terms in the numerator. Divide out
common factors.
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Simplify.
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Check It Out! Example 1b
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.
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Additional Example 2 Subtracting Rational
Expressions with Like Denominators
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.
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Check It Out! Example 2a
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.
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Check It Out! Example 2b
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. There are no common factors.
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As with fractions, rational expressions must have
a common denominator before they can be added or
subtracted. If they do not have a common
denominator, you can use any common multiple of
the denominators to find one. You can also use
the least common multiple (LCM) of the
denominators.
To find the LCM of two expressions, write the
prime factorization of both expressions. Line up
the factors as shown. To find the LCM, multiply
one number from each column.
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Additional Example 3A Identifying the Least
Common Multiple
Find the LCM of the given expressions.
12x2y, 9xy3
Write the prime factorization of each expression.
Align common factors.
9xy3 3 ? 3 ? x ? y ? y ? y
12x2y 2 ? 2 ? 3 ? x ? x ? y?
LCM 2 ? 2 ? 3 ? 3 ? x ? x ? y ? y ? y
36x2y3
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Additional Example 3B Identifying the Least
Common Multiple
Find the LCM of the given expressions.
c2 8c 15, 3c2 18c 27
Factor each expression.
3c2 18c 27 3(c2 6c 9)
3(c 3)(c 3)
Align common factors.
c2 8c 15 (c 3) (c 5)
LCM 3(c 3)2(c 5)
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Check It Out! Example 3a
Find the LCM of the given expressions.
5f2h, 15fh2
Write the prime factorization of each expression.
Align common factors.
5f2h 5 ? f ? f ? h
15fh2 3 ? 5 ? f ? h ? h
LCM 3 ? 5 ? f ? f ? h ? h
15f2h2
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Check It Out! Example 3b
Find the LCM of the given expressions.
x2 4x 12, (x 6)(x 5)
Factor each expression.
x2 4x 12 (x 6) (x 2)
Align common factors.
(x 6)(x 5) (x 6)(x 5)
LCM (x 6)(x 5)(x 2)
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The LCM of the denominators of rational
expressions is also called the least common
denominator, or LCD, of the rational expressions.
You can use the LCD to add or subtract rational
expressions.
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Adding or Subtracting Rational Expressions
Step 1 Identify a common denominator.
Step 2 Multiply each expression by an appropriate
form of 1 so that each term has the common
denominator as its denominator.
Step 3 Write each expression using the common
denominator.
Step 4 Add or subtract the numerators, combining
like terms as needed.
Step 5 Factor as needed.
Step 6 Simplify as needed.
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Additional Example 4A Adding and Subtracting
with Unlike Denominators
Add or subtract. Simplify your answer.
Identify the LCD.
5n3 5 ? n ? n ? n
Step 1
2n2 2 ? n ? n
LCD 2 ? 5 ? n ? n ? n 10n3
Multiply each expression by an appropriate form
of 1.
Step 2
Write each expression using the LCD.
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Additional Example 4A Continued
Add or subtract. Simplify your answer.
Add the numerators.
Factor and divide out common factors.
Step 5
Simplify.
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Additional Example 4B Adding and Subtracting
with Unlike Denominators.
Add or subtract. Simplify your answer.
Step 1 The denominators are opposite binomials.
The LCD can be either w 5 or 5 w.
Identify the LCD.
Write each expression using the LCD.
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Additional Example 4B Continued
Add or Subtract. Simplify your answer.
Subtract the numerators.
No factoring needed, so just simplify.
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Check It Out! Example 4a
Add or subtract. Simplify your answer.
Identify the LCD.
Multiply each expression by an appropriate form
of 1.
Write each expression using the LCD.
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Check It Out! Example 4a Continued
Add or subtract. Simplify your answer.
Subtract the numerators.
Step 5
Factor and divide out common factors.
Simplify.
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Check It Out! Example 4b
Add or subtract. Simplify your answer.
Factor the first term. The denominator of second
term is a factor of the first.
Add the two fractions.
Divide out common factors.
Step 4
Simplify.
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Additional Example 5 Recreation Application
Roland needs to take supplies by canoe to some
friends camping 2 miles upriver and then return
to his own campsite. Rolands average paddling
rate is about twice the speed of the rivers
current.
a. Write and simplify an expression for how long
it will take Roland to canoe round trip.
Step 1 Write expressions for the distances and
rates in the problem. The distance in both
directions is 2 miles.
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Additional Example 5 Continued
Let x represent the rate of the current, and let
2x represent Rolands paddling rate.
Rolands rate against the current is 2x x, or
x. Rolands rate with the current is 2x x, or
3x.
Step 2 Use a table to write expressions for time.
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Additional Example 5 Continued
Step 3 Write and simplify an expression for the
total time.
total time time upstream time downstream
Substitute known values.
Multiply the first fraction by an appropriate
form of 1.
Write each expression using the LCD, 3x.
Add the numerators.
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Additional Example 5 Continued
b. The speed of the rivers current is 2.5 miles
per hour. About how long will it take Roland to
make the round trip?
Substitute 2.5 for x. Simplify.
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Check It Out! Example 5
What if?...Katys average paddling rate increases
to 5 times the speed of the current. Now how long
will it take Katy to kayak the round trip?
Step 1 Let x represent the rate of the current,
and let 5x represent Katys paddling rate.
Katys rate against the current is 5x x, or 4x.
Katys rate with the current is 5x x, or 6x.
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Check It Out! Example 5 Continued
Step 2 Use a table to write expressions for time.
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Check It Out! Example 5 Continued
Step 3 Write and simplify an expression for the
total time.
total time time upstream time downstream
Substitute known values.
Multiply each fraction by an appropriate form of
1.
Write each expression using the LCD, 12x.
Add the numerators.
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Check It Out! Example 5 Continued
b. If the speed of the rivers current is 2 miles
per hour, about how long will it take Katy to
make the round trip?
Substitute 2 for x. Simplify.
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Lesson Quiz Part I
Add or subtract. Simplify your answer.
1.
2.
3.
4.
5.
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Lesson Quiz Part II
6. Vong drove 98 miles on interstate highways and
80 miles on state roads. He drove 25 faster on
the interstate highways than on the state roads.
Let r represent his rate on the state roads in
miles per hour.
a. Write and simplify an expression that
represents the number of hours Vong drove in
terms of r.
b. Find Vongs driving time if he averaged 55
miles per hour on the state roads.
about 2 h 53 min