NAME
DATE
11-8
PERIOD
Study Guide and Intervention
Rational Equations
Solve Rational Equations Rational equations are equations that contain rational
expressions. To solve equations containing rational expressions, multiply each side of the
equation by the least common denominator.
Rational equations can be used to solve work problems and rate problems.
x-3
x
Solve −
+−
= 4.
x-3
x
−
+−
=4
3
2
x-3
x
6 − + − = 6(4)
3
2
(
)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2(x - 3) + 3x = 24
2x - 6 + 3x = 24
5x = 30
x=6
The solution is 6.
3
15
5
= −
. State
Solve −
2
Example 2
2
2(x - 1)
x -1
any extraneous solutions.
15
5
−
=−
2
Distributive Property
Distributive Property
Simplify.
Divide each side by 5.
Original equation
2(x - 1)
x -1
The LCD is 6.
30(x - 1) = 5(x2 - 1)
30x - 30 = 5x2 - 5
0 = 5x2 - 30x + 30 - 5
Cross multiply.
Distributive Property
Add -30x + 30 to
each side.
2
0 = 5x - 30x + 25
Simplify.
2
0 = 5(x - 6x + 5)
Factor.
0 = 5(x - 1)(x - 5)
Factor.
x = 1 or x = 5
Zero Product Property
The number 1 is an extraneous solution, since 1
is an excluded value for x. So, 5 is the solution of
the equation.
Exercises
Solve each equation. State any extraneous solutions.
x-5
x
1. −
+−
= 8 20
3
6
2. −
x =− 1
8
10
4. −
=−
9
4
5. t - −
= t + 3 -4−
5
4
n-1
n+1
q+4
q-1
q
q+1
3
2
m
= 1 -2
9. − - −
1-m
2
1
11. −
-−
= 0 -4; 6 is
2
x - 36
x-6
p2
p-4
extraneous
4
13. −
- − = 4 -6, 2
4-p
Chapter 11
5
1
3
t+3
7. − + − = 2 - −
m+1
m-1
x-1
2x - 2
3. −
=−
1
x+1
15
m+4
m
m
6. −
m + − = − -4; 0 is
3
3
extraneous
4x + 3
7x + 2 10
5 - 2x
8. −
-−=− −
2
6
6
17
x2 - 9
10. −
+ x2 = 9 -3 or 2
x-3
4z
6
4
12. −
=−
+−
-3 is
2
z + 4z + 3
z+3
z+1
extraneous
x2 - 16
14. −
+ x 2 = 16 -4, 3; 4 is
x-4
49
extraneous
Glencoe Algebra 1
Lesson 11-8
Example 1
NAME
DATE
11-8
Study Guide and Intervention
PERIOD
(continued)
Rational Equations
Use Rational Equations to Solve Problems
Rational equation can be used to
solve work problems and rate problems.
Example
WORK PROBLEM Marla can paint Percy’s kitchen in 3 hours. Percy
can paint it in 2 hours. Working together, how long will it take Marla and Percy to
paint the kitchen?
1
1
of the job and Percy completes t · −
of the job. So an
In t hours, Marla completes t · −
3
equation for completing the whole job is −t + −t = 1.
3
−t + −t = 1
3
2
2t + 3t = 6
5t = 6
6
t=−
5
2
2
Multiply each term by 6.
Add like terms.
Solve.
1
So it will take Marla and Percy 1−
hours to paint the room if they work together.
5
Exercises
1. GREETING CARDS It takes Kenesha 45 minutes to prepare 20 greeting cards. It takes
Paula 30 minutes to prepare the same number of cards. Working together at this rate,
how long will it take them to prepare the cards? 18 min
3. FLOORING Maya and Reginald are installing hardwood flooring. Maya can install
flooring in a room in 4 hours. Reginald can install flooring in a room in 3 hours. How
12
long would it take them if they worked together? −
h or about 1.71 hours
7
4. BICYCLING Stefan is bicyling on a bike trail at an average of 10 miles per hour. Erik
starts bicycling on the same trail 30 minutes later. If Erik averages 16 miles per hour,
how long will it take him to pass Stefan? 50 min
Chapter 11
50
Glencoe Algebra 1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. BOATING A motorboat went upstream at 15 miles per hour and returned downstream
at 20 miles per hour. How far did the boat travel one way if the round trip took
3.5 hours? 30 mi