NAME ______________________________________________ DATE
____________ PERIOD _____
12-9 Solving Rational Equations
(Pages 690–695)
A rational equation is an equation that contains rational expressions. To
solve a rational equation, multiply each side of the equation by the LCD of
the rational expressions in the equation. Doing so can yield results that are
not solutions to the original equation, called extraneous solutions or
“false” solutions. To eliminate extraneous solutions, be sure no solution is an
excluded value of the original equation.
Example
a
3a 4
Solve 3.
a1
a1
3a 4
a
3
a1
a1
3a 4
(a 1)a (a 1)3
a1
a1
3a 4
(a 1)a (a 1)
(a 1)3
a1
a1
a 3a 4 3a 3
4a 4 3a 3
a43
Multiply each side by the LCD, a 1.
Use the Distributive Property.
Multiply.
Add.
Subtract 3a from each side.
a 1
Subtract 4 from each side.
Since 1 is an excluded value of the original equation, 1 is an extraneous solution.
Thus, this equation has no solution.
Practice
Solve each equation.
3
3. 7 2x
1. 2 4 1
2. n 4 5
4. 1 3
x2
9
5. (x 7) x
6. 2x 4
a1
a1
8. 2n
n3
9. 2
3y
y
n
3
t6
t
x
k8
k4
7. 3
k
x
k
x
a4
a
n1
w5
w
1
10. 11. x 1
13. x 2 x
14. 1 w6
8
x
4
4
4
x2
x2
n1
n1
n1
12. x
n
y3
y2
y1
y2
n3
c
c4
15. 3 c2
4
16. Standardized Test Practice Solve x 1 2.
3
A x 3
x
x
B x3
C x 3, 3
D no solution
13. 4, 4 14. no solution 15. 10, 4 16. C
4. 3 5. 7, 1 6. 0, 4 7. 4 8. 1 9. no solution 10. 7, 2 11. 1, 2 12. 3
99
2
Glencoe/McGraw-Hill
Answers: 1. 14 2. 1, 5 3. 3, 1
©
Glencoe Algebra 1