# 12.7

```12-7Solving
12-7
SolvingRational
RationalEquations
Equations
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
11
12-7 Solving Rational Equations
Warm Up
1. Find the LCM of x, 2x2, and 6.
2. Find the LCM of p2 – 4p and p2 – 16.
3.
5.
Holt Algebra 1
4.
12-7 Solving Rational Equations
Objectives
Solve rational equations.
Identify extraneous solutions.
Holt Algebra 1
12-7 Solving Rational Equations
Vocabulary
rational equation
Holt Algebra 1
12-7 Solving Rational Equations
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.
Holt Algebra 1
12-7 Solving Rational Equations
Example 1: Solving Rational Equations by Using
Cross Products
Solve
Use cross products.
5x = (x – 2)(3)
5x = 3x – 6
2x = –6
x = –3
Check
Distribute 3 on the
right side.
Subtract 3x from
both sides.
–1 –1 
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 1a
Solve
Check
Use cross products.
3n = (n + 4)(1)
3n = n + 4
2n = 4
n=2
Holt Algebra 1
Distribute 1 on the
right side.
Subtract n from both sides.
Divide both sides by 2.

12-7 Solving Rational Equations
Check It Out! Example 1b
Solve
Check
Use cross products.
4h = (h + 1)(2)
4h = 2h + 2
Distribute 2 on the
right side.
2h = 2
Subtract 2h from both sides.
h=1
Divide both sides by 2.
Holt Algebra 1

12-7 Solving Rational Equations
Check It Out! Example 1c
Solve
Check
Use cross products.
21x = (x – 7)(3)
21x = 3x –21
18x = –21
x=
Holt Algebra 1
Distribute 3 on the
right side.
Subtract 3x from both sides.
Divide both sides by 18.

12-7 Solving Rational Equations
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.
Holt Algebra 1
12-7 Solving Rational Equations
Example 2A: Solving Rational Equations by Using the
LCD
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.
Holt Algebra 1
12-7 Solving Rational Equations
Example 2A Continued
Step 3 Simplify and solve.
Divide out common factors.
(2x)(2) +6(x +1) = 5(x +1)
4x + 6x + 6 = 5x + 5
10x + 6 = 5x + 5
5x = –1
Holt Algebra 1
Simplify.
Distribute and multiply.
Combine like terms.
Subtract 5x and 6
from both sides.
Divide both sides by 5.
12-7 Solving Rational Equations
Example 2A Continued
Check Verify
that your
solution is not
extraneous.

Holt Algebra 1
12-7 Solving Rational Equations
Example 2B: Solving Rational Equations by Using the
LCD
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.
Holt Algebra 1
12-7 Solving Rational Equations
Example 2B Continued
Step 3 Simplify and solve.
Divide out common
factors.
4x – 3 = x2
–x2 + 4x – 3 = 0
x2 – 4x + 3 = 0
(x – 3)(x – 1) = 0
Simplify.
Subtract x2 from both
sides.
Multiply by – 1.
Factor.
x = 3 or 1 Solve.
Holt Algebra 1
12-7 Solving Rational Equations
Example 2B Continued
Check Verify that your solution is not extraneous.


Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2a
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.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2a Continued
Step 3 Simplify and solve.
Divide out
common
factors.
3a = 4(a + 1) Simplify.
3a = 4a + 4
–4 = a
Holt Algebra 1
Distribute the 4.
Subtract the 4 and 3a
from both sides.
12-7 Solving Rational Equations
Check It Out! Example 2a Continued
Check Verify that your solution is not extraneous.

Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2b
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.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2b Continued
Divide out common
terms.
12j – 10(2j + 4) = 4j + 8
12j – 20j – 40 = 4j + 8
–12j = 48
j = –4
Holt Algebra 1
Simplify.
Distribute 10.
Combine like terms.
12-7 Solving Rational Equations
Check It Out! Example 2b Continued
Check Verify that your solution is not extraneous.

Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2c
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.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 2c Continued
Divide out
common terms.
8t = (t + 3) + t(t + 3)
8t = t + 3 + t2 + 3t
0 = t2 – 4t + 3
Distribute t.
Combine like terms.
0 = (t – 3)(t – 1)
Factor.
t = 3, 1
Holt Algebra 1
Simplify.
12-7 Solving Rational Equations
Check It Out! Example 2c Continued
Check Verify that your solution is not extraneous.

Holt Algebra 1

12-7 Solving Rational Equations
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?
Holt Algebra 1
12-7 Solving Rational Equations
1
Understand the Problem
The answer will be the number of minutes
m machine A and machine B need to print
the copies.
List the important information:
• Machine A can print the copies in 60
minutes, which is
of the job in 1 minute.
• Machine B can print the copies in 10
minutes, which is
of the job in 1 minute.
Holt Algebra 1
12-7 Solving Rational Equations
2
Make a Plan
The part of the copies that machine A can
print plus the part that machine B can print
equals the complete job. Machine A’s rate
times the number of minutes plus machine
B’s rate times the number of minutes will
give the complete time to print the copies.
(machine
A’s rate)
m
Holt Algebra 1
m + (machine
B’s rate)
+
m
m = complete
job
=
1
12-7 Solving Rational Equations
3
Solve
Multiply both sides by the
LCD, 60.
1m + 6m = 60
7m = 60
Distribute 60 on the left
side.
Combine like terms.
Divide both sides by 7.
Machine A and Machine B working together can
print the copies in a little more than 8.5 minutes.
Holt Algebra 1
12-7 Solving Rational Equations
4 Look Back
Machine A prints
of the copies per minute
and machine B prints
of the copies per
minute. So in
minutes, machine A prints
of the copies and machine B prints
of the copies. Together, they print
Holt Algebra 1
12-7 Solving Rational Equations
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?
1
Understand the Problem
The answer will be the number of minutes
m Sara and Cindy need to mow the lawn.
Holt Algebra 1
12-7 Solving Rational Equations
1
Understand the Problem
The answer will be the number of minutes m
Sara and Cindy need to mow the lawn.
List the important information:
• Cindy can mow the lawn in 50 minutes,
which is
of the job in 1 minute.
• Sara can mow the lawn in 40 minutes,
which is
of the job in 1 minute.
Holt Algebra 1
12-7 Solving Rational Equations
2
Make a Plan
The part of the lawn Cindy can mow plus the
part of the lawn Sara can mow equals the
complete job. Cindy’s rate times the number
of minutes plus Sara’s rate times the number
of minutes will give the complete time to
mow the lawn.
(Cindy’s rate) m + (Sara’s rate) m = lawn mowed
m
Holt Algebra 1
+
m
=
1
12-7 Solving Rational Equations
3
Solve
Multiply both sides by the
LCD, 200.
5m + 4m = 200
9m = 200
Distribute 200 on the left
side.
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.
Holt Algebra 1
12-7 Solving Rational Equations
4
Look Back
Cindy mows
mows
of the lawn in 1 minute and Sara
of the lawn in 1 minute. So, in
minutes Cindy mows
Sara mows
mow
Holt Algebra 1
of the lawn and
of the lawn. Together they
lawn.
12-7 Solving Rational Equations
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.
Holt Algebra 1
12-7 Solving Rational Equations
Extraneous solutions may be introduced by
squaring both sides of an equation or by
multiplying both sides of an equation by a
variable expression.
Holt Algebra 1
12-7 Solving Rational Equations
Example 4: Extraneous Solutions
Solve
solutions.
Step 1 Solve.
. Identify any extraneous
Use cross products.
2(x2 – 1) = (x + 1)(x – 6) Distribute 2 on the left side.
Multiply the right side.
2x2 – 2 = x2 – 5x – 6
Subtract x2 from both sides.
Add 5x and 6 to both sides.
x2 + 5x + 4 = 0
(x + 1)(x + 4) = 0
Use the Zero Product Property.
Solve.
x = –1 or x = –4
Holt Algebra 1
12-7 Solving Rational Equations
Example 4 Continued
Solve
solutions.
. Identify any extraneous
Step 2 Find extraneous solutions.

Because
and
are undefined –1 is
not a solution.

The only solution is – 4, so – 1 is an extraneous solution.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 4a
Solve. Identify any extraneous solutions.
Step 1 Solve.
Use cross products.
(x – 2)(x – 7) = 3(x – 7) Distribute 3 on the right side.
Multiply the left side.
2x2 – 9x + 14 = 3x – 21 Subtract 3x from both sides.
X2 – 12x + 35 = 0
(x – 7)(x – 5) = 0
Use the Zero Product Property.
Solve.
x = 7 or x = 5
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 4a Continued
Step 2 Find extraneous solutions.

Because
and
are undefined 7 is
not a solution.

The only solution is 5, so 7 is an extraneous solution.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 4b
Solve. Identify any extraneous solutions.
Step 1 Solve.
(x + 1)(x – 3) = 4(x – 2)
x2 – 2x – 3 = 4x – 8
X2 – 6x + 5 = 0
(x – 1)(x – 5) = 0
x = 1 or x = 5
Holt Algebra 1
Use cross products.
Distribute 4 on the right side.
Multiply the left side.
Subtract 4x from both sides.
Use the Zero Product Property.
Solve.
12-7 Solving Rational Equations
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.
Holt Algebra 1
12-7 Solving Rational Equations
Check It Out! Example 4c
Solve. Identify any extraneous solutions.
Step 1 Solve.
6(x2 + 2x) = 9(x2)
3x2 – 12x = 0
3x(x – 4) = 0
3x = 0, or x – 4 = 0
x = 0 or x = 4
Holt Algebra 1
Use cross products.
Distribute 6 on the left side.
Multiply the right side.
Subtract 9x2 from both sides.
Multiply through with – 1.
Use the Zero Product Property.
Solve.
12-7 Solving Rational Equations
Check It Out! Example 4c Continued
Step 2 Find extraneous solutions.

Because
and
are undefined 0 is
not a solution.

The only solution is 4, so 0 is an extraneous solutions.
Holt Algebra 1
12-7 Solving Rational Equations
Lesson Quiz: Part I
1.
24
2.
–4, 3
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?
Holt Algebra 1
12-7 Solving Rational Equations
Lesson Quiz: Part II
5. Solve
.
Identify any extraneous solutions.
–5; 3 is extraneous.
Holt Algebra 1
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