10-5 Adding and Subtracting Rational Expressions Warm Up Warm Up Lesson Presentation Lesson...
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Transcript of 10-5 Adding and Subtracting Rational Expressions Warm Up Warm Up Lesson Presentation Lesson...
10-5 Adding and Subtracting Rational Expressions
Warm UpWarm Up
Lesson Presentation
California Standards
PreviewPreview
10-5 Adding and Subtracting Rational Expressions
Warm UpAdd. Simplify your answer.
1. 2.
3. 4.
Subtract. Simplify your answer.
5.
7.
6.
8.
10-5 Adding and Subtracting Rational Expressions
13.0 Students add, subtract, multiply and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Also covered: 15.0
California Standards
10-5 Adding and Subtracting Rational Expressions
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.
10-5 Adding and Subtracting Rational Expressions
Additional Example 1A: Adding Rational Expressions with Like Denominators
Add. Simplify your answer.
Combine like terms in the numerator. Divide out common factors.
Simplify.
10-5 Adding and Subtracting Rational Expressions
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.
10-5 Adding and Subtracting Rational Expressions
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.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 1a
Add. Simplify your answer.
= 2
Combine like terms in the numerator. Divide out common factors.
Simplify.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 1b
Add. Simplify your answer.
Combine like terms in the numerator.
Factor. Divide out common factors.
Simplify.
10-5 Adding and Subtracting Rational ExpressionsAdditional Example 2: Subtracting Rational Expressions
with Like DenominatorsSubtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.
10-5 Adding and Subtracting Rational Expressions
Make sure you add the opposite of each term in the numerator of the second expression when subtracting rational expressions.
Caution
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 2a
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. Divide out common factors.
Simplify.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 2b
Subtract. Simplify your answer.
Subtract numerators.
Combine like terms.
Factor. There are no common factors.
10-5 Adding and Subtracting Rational Expressions
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.
10-5 Adding and Subtracting Rational Expressions
Additional Example 3A: Identifying the Least Common Multiple
Find the LCM of the given expressions.
12x2y, 9xy3
12x2y = 2 2 3 x x y
9xy3 = 3 3 x y y y
LCM = 2 2 3 3 x x y y y
Write the prime factorization of each expression. Align common factors. = 36x2y3
10-5 Adding and Subtracting Rational Expressions
Additional Example 3B: Identifying the Least Common Multiple
Find the LCM of the given expressions.
c2 + 8c + 15, 3c2 + 18c + 27
c2 + 8c + 15 = (c + 3) (c + 5)
3c2 + 18c + 27 = 3(c2 + 6c +9)
= 3(c + 3)(c + 3)
LCM = 3(c + 3)2(c + 5)
Factor each expression.
Align common factors.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 3a
Find the LCM of the given expressions.
5f2h, 15fh2
5f2h = 5 f f h
15fh2 = 3 5 f h h
LCM = 3 5 f f h h
= 15f2h2
Write the prime factorization of each expression. Align common factors.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 3b
Find the LCM of the given expressions.
x2 – 4x – 12, (x – 6)(x + 5)
x2 – 4x – 12 = (x – 6) (x + 2)
(x – 6)(x + 5) = (x – 6)(x + 5)
LCM = (x – 6)(x + 5)(x + 2)
Factor each expression.
Align common factors.
10-5 Adding and Subtracting Rational Expressions
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.
10-5 Adding and Subtracting Rational Expressions
Adding or Subtracting Rational Expressions
Step 1 Identify a common denominator.
Step 3 Write each expression using the 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 4 Add or subtract the numerators, combining like terms as needed.
Step 5 Factor as needed.
Step 6 Simplify as needed.
10-5 Adding and Subtracting Rational Expressions
Additional Example 4A: Adding and Subtracting with Unlike Denominators
Add or subtract. Simplify your answer.
Step 15n3 = 5 n n n2n2 = 2 n nLCD = 2 5 n n n = 10n3
Identify the LCD.
Step 2Multiply each expression
by an appropriate form of 1.
Write each expression using the LCD.
Step 3
10-5 Adding and Subtracting Rational Expressions
Additional Example 4A Continued
Add or subtract. Simplify your answer.
Add the numerators.
Factor and divide out common factors.
Step 6 Simplify.
Step 4
Step 5
10-5 Adding and Subtracting Rational Expressions
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.
Step 2
Step 3
Multiply the first expression
by to get an LCD of
w – 5. Write each expression
using the LCD.
10-5 Adding and Subtracting Rational Expressions
Additional Example 4B Continued
Add or Subtract. Simplify your answer.
Step 4
Step 5, 6
Subtract the numerators.
No factoring needed, so just simplify.
10-5 Adding and Subtracting Rational Expressions
Add or subtract. Simplify your answer.
Identify the LCD.3d 3 d 2d3 = 2 d d d
LCD = 2 3 d d d = 6d3 Step 1
Multiply each expression by an appropriate form of 1.
Write each expression using the LCD.
Check It Out! Example 4a
Step 2
Step 3
10-5 Adding and Subtracting Rational Expressions
Add or subtract. Simplify your answer.
Check It Out! Example 4a Continued
Subtract the numerators.
Factor and divide out common factors.
Step 4
Simplify.
Step 5
Step 6
10-5 Adding and Subtracting Rational Expressions
Add or subtract. Simplify your answer.Check It Out! Example 4b
Factor the first term. The denominator of second term is a factor of the first.
Add the two fractions.
Divide out common factors.
Step 1
Step 4 Simplify.
Step 2
Step 3
10-5 Adding and Subtracting Rational Expressions
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. Roland’s average paddling rate is about twice the speed of the river’s 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.
10-5 Adding and Subtracting Rational Expressions
Additional Example 5 Continued
Roland’s rate against the current is 2x – x, or x. Roland’s rate with the current is 2x + x, or 3x.
Step 2 Use a table to write expressions for time.
Downstream
(with current)
Upstream
(against current)
Rate
(mi/h)
Distance (mi)
DirectionTime (h) = Distance
rate
2
2
x
3x
Let x represent the rate of the current, and let 2x represent Roland’s paddling rate.
10-5 Adding and Subtracting Rational Expressions
Additional Example 5 Continued
Step 3 Write and simplify an expression for the total time.
total time = time upstream + time downstream
total time = Substitute known values.
Multiply the first fraction by an appropriate form of 1.
Write each expression using the LCD, 3x.
Add the numerators.
Step 4
Step 5
Step 6
10-5 Adding and Subtracting Rational Expressions
Additional Example 5 Continued
b. The speed of the river’s 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.
It will take Roland of an hour or 64 minutes to make the round trip.
10-5 Adding and Subtracting Rational Expressions
Check It Out! Example 5
What if?...Katy’s 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 Katy’s paddling rate.
Katy’s rate against the current is 5x – x, or 4x. Katy’s rate with the current is 5x + x, or 6x.
10-5 Adding and Subtracting Rational Expressions
Step 2 Use a table to write expressions for time.
Check It Out! Example 5 Continued
Downstream
(with current)
Upstream
(against current)
Rate
(mi/h)
Distance (mi)
DirectionTime (h) = distance
rate
1
1
4x
6x
10-5 Adding and Subtracting Rational Expressions
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.
total time =
Step 4
Step 5
Step 6
10-5 Adding and Subtracting Rational Expressions
b. If the speed of the river’s current is 2 miles per hour, about how long will it take Katy to make the round trip?
Substitute 2 for x. Simplify.
Check It Out! Example 5 Continued
It will take Katy of an hour or 12.5 minutes to make the round trip.
10-5 Adding and Subtracting Rational Expressions
Lesson Quiz: Part I
Add or subtract. Simplify your answer.
1. 2.
5.
3. 4.
10-5 Adding and Subtracting Rational Expressions
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 Vong’s driving time if he averaged 55 miles per hour on the state roads.about 2 h 53 min