Solving Rational Equations

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12-7
Solving Rational Equations
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1
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• Warm Up
• 1. Find the LCM of x, 2x2, and 6.
• 2. Find the LCM of p2 4p and p2 16.
• Multiply. Simplify your answer.
• 3. 4.

5.
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Objectives
Solve rational equations. Identify extraneous
solutions.
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Vocabulary
rational equation
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A rational equation is an equation that contains
one or more rational expressions. If a rational
equation is a proportion, it can be solved using
the Cross Product Property.
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Example 1 Solving Rational Equations by Using
Cross Products
Check
Use cross products.
5x (x 2)(3)
Distribute 3 on the right side.
5x 3x 6
2x 6
Subtract 3x from both sides.
x 3
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Check It Out! Example 1a
Use cross products.
3n (n 4)(1)
Distribute 1 on the right side.
3n n 4
Subtract n from both sides.
2n 4
n 2
Divide both sides by 2.
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Check It Out! Example 1b
Solve . Check your answer.
Use cross products.
4h (h 1)(2)
Distribute 2 on the right side.
4h 2h 2
2h 2
Subtract 2h from both sides.
h 1
Divide both sides by 2.
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Check It Out! Example 1c
Solve . Check your answer.
Use cross products.
21x (x 7)(3)
Distribute 3 on the right side.
21x 3x 21
Subtract 3x from both sides.
18x 21
?
x
Divide both sides by 18.
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Some rational equations contain sums or
differences of rational expressions. To solve
these, you must find the LCD of all the rational
expressions in the equation.
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Example 2A Solving Rational Equations by Using
the LCD
Solve each equation. Check your answer.
Step 1 Find the LCD
2x(x 1)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD
Distribute on the left side.
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Example 2A Continued
Step 3 Simplify and solve.
Divide out common factors.
(2x)(2) 6(x 1) 5(x 1)
Simplify.
4x 6x 6 5x 5
Distribute and multiply.
10x 6 5x 5
Combine like terms.
Subtract 5x and 6 from both sides.
5x 1
Divide both sides by 5.
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Example 2A Continued
Check Verify that your solution is not extraneous.
?
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Example 2B Solving Rational Equations by Using
the LCD
Solve each equation. Check your answer.
Step 1 Find the LCD
(x2)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD
Distribute on the left side.
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Example 2B Continued
Step 3 Simplify and solve.
Divide out common factors.
4x 3 x2
Simplify.
Subtract x2 from both sides.
x2 4x 3 0
x2 4x 3 0
Multiply by 1.
(x 3)(x 1) 0
Factor.
x 3, 1
Solve.
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Example 2B Continued
Check Verify that your solution is not extraneous.
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Check It Out! Example 2a
Solve each equation. Check your answer.
Step 1 Find the LCD
a(a 1)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD
Distribute on the left side.
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Check It Out! Example 2a Continued
Step 3 Simplify and solve.
Divide out common factors.
3a 4(a 1)
Simplify.
3a 4a 4
Distribute the 4.
4 a
Subtract the 4 and 3a from both sides.
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Check It Out! Example 2a Continued
Check Verify that your solution is not extraneous.
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Check It Out! Example 2b
Solve each equation. Check your answer.
Step 1 Find the LCD
2j(j 2)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD
Distribute on the left side.
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Check It Out! Example 2b Continued
Solve each equation. Check your answer.
Divide out common terms.
12j 10(2j 4) 4j 8
Simplify.
12j 20j 40 4j 8
Distribute 10.
12j 48
Combine like terms.
j 4
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Check It Out! Example 2b Continued
Check Verify that your solution is not extraneous.
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Check It Out! Example 2c
Solve each equation. Check your answer.
Step 1 Find the LCD
t(t 3)
Include every factor of the denominator.
Step 2 Multiply both sides by the LCD
Distribute on the right side.
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Check It Out! Example 2c Continued
Solve each equation. Check your answer.
Divide out common terms.
8t (t 3) t(t 3)
Simplify.
8t t 3 t2 3t
Distribute t.
0 t2 4t 3
Combine like terms.
0 (t 3)(t 1)
Factor.
t 3, 1
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Check It Out! Example 2c Continued
Check Verify that your solution is not extraneous.
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Example 3 Problem-Solving Application
Copy machine A can make 200 copies in 60 minutes.
Copy machine B can make 200 copies in 10 minutes.
How long will it take both machines working
together to make 200 copies?
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The answer will be the number of minutes m
machine A and machine B need to print the copies.
List the important information
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The part of the copies that machine A can print
plus the part that machine B can print equals the
complete job. Machine As rate times the number
of minutes plus machine Bs rate times the number
of minutes will give the complete time to print
the copies.
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Multiply both sides by the LCD, 60.
Distribute 60 on the left side.
1m 6m 60
Combine like terms.
7m 60
Divide both sides by 7.
Machine A and Machine B working together can
print the copies in a little more than 8.5
minutes.
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Look Back
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Check It Out! Example 3
Cindy mows a lawn in 50 minutes. It takes Sara 40
minutes to mow the same lawn. How long will it
take them to mow the lawn if they work together?
The answer will be the number of minutes m Sara
and Cindy need to mow the lawn.
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The answer will be the number of minutes m Sara
and Cindy need to mow the lawn.
List the important information
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The part of the lawn Cindy can mow plus the part
of the lawn Sara can mow equals the complete
job. Cindys rate times the number of minutes
plus Saras rate times the number of minutes will
give the complete time to mow the lawn.
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Multiply both sides by the LCD, 200.
Distribute 200 on the left side.
5m 4m 200
9m 200
Combine like terms.
Divide both sides by 9.
Cindy and Sara working together can mow the lawn
in a little more than 22.2 minutes.
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Look Back
Cindy mows of the lawn in 1 minute and Sara
mows of the lawn in 1 minute. So, in
minutes Cindy mows of the lawn
and Sara mows of the lawn.
Together they mow lawn.
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When you multiply each side of an equation by the
LCD, you may get an extraneous solution. Recall
from Chapter 11 that an extraneous solution is a
solution to a resulting equation that is not a
solution to the original equation.
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Example 4 Extraneous Solutions
Solve . Identify any
extraneous solutions.
Step 1 Solve.
Use cross products.
Distribute 2 on the left side. Multiply the right
side.
2(x2 1) (x 1)(x 6)
2x2 2 x2 5x 6
Subtract x2 from both sides. Add 5x and 6 to both
sides.
x2 5x 4 0
Factor the quadratic expression.
(x 1)(x 4) 0
Use the Zero Product Property.
Solve.
x 1 or x 4
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Example 4 Continued
Solve . Identify any
extraneous solutions.
Step 2 Find extraneous solutions.
The only solution is 4, so 1 is an extraneous
solution.
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Check It Out! Example 4a
Solve. Identify any extraneous solutions.
Step 1 Solve.
Use cross products.
Distribute 3 on the right side. Multiply the left
side.
(x 2)(x 7) 3(x 7)
2x2 9x 14 3x 21
Subtract 3x from both sides. Add 21 to both sides.
X2 12x 35 0
Factor the quadratic expression.
(x 7)(x 5) 0
Use the Zero Product Property.
Solve.
x 7 or x 5
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Check It Out! Example 4a Continued
Step 2 Find extraneous solutions.
The only solution is 5, so 7 is an extraneous
solution.
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Check It Out! Example 4b
Solve. Identify any extraneous solutions.
Step 1 Solve.
Use cross products.
Distribute 4 on the right side. Multiply the left
side.
(x 1)(x 3) 4(x 2)
x2 2x 3 4x 8
Subtract 4x from both sides. Add 8 to both sides.
X2 6x 5 0
Factor the quadratic expression.
(x 1)(x 5) 0
Use the Zero Product Property.
Solve.
x 1 or x 5
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Check It Out! Example 4b Continued
Step 2 Find extraneous solutions.
1 and 5 are solutions.
The solutions are 1 and 5, there are no
extraneous solutions.
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Check It Out! Example 4c
Solve. Identify any extraneous solutions.
Step 1 Solve.
Use cross products.
Distribute 6 on the left side. Multiply the right
side.
6(x2 2x) 9(x2)
3x2 12x 0
3x(x 4) 0
Factor the quadratic expression.
3x 0, or x 4 0
Use the Zero Product Property.
Solve.
x 0 or x 4
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Check It Out! Example 4c Continued
Step 2 Find extraneous solutions.
The only solution is 4, so 0 is an extraneous
solutions.
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Lesson Quiz Part I
Solve each equation. Check your answer.
4, 3
2.
1.
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3.
4. Pipe A can fill a tank with water in 4 hours.
Pipe B can fill the same tank in 5 hours. How
long will it take both pipes working together to
fill the tank?
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Lesson Quiz Part II
5. Solve
. Identify any extraneous solutions.
5 3 is extraneous.