Notes on Solving Rational Equations - Page 1
Name_________________________
A rational expression is the ratio (or division) of two polynomials. Here is a more basic way of
stating the definition of a rational expression: "a fraction with variables in it".
3 3y  7
z
x 2  10
The following are examples of rational expressions:
,
,
,
.
x
2
z  5 y4  8
--------------------------------------------------------------------------------------------------------------------A rational equation is an equation with one or more rational expressions.
The objective of this unit is to solve rational equations.
a c
 . In other words, we will learn how to solve a
b d
rational equation in which two rational expressions have been set equal to each other. Rational
equations in this form are known as proportions.
---------------------------------------------------------------------------------------------------------------------
The first type to be learned is of the form
How to Solve Proportions of the Form
a c

b d
1.
Place parentheses around any numerator or denominator with more than one term.
2.
Cross multiply and simplify. This means rewriting the equation as a  d  b  c .
3.
A)If the resulting equation is linear (the highest exponent of the variable is 1), then
isolate the variable.
B)If the resulting equation is quadratic (the highest exponent of the variable is 2), then
solve the equation using the techniques previously learned.
4.
Check your answer. Occasionally, you will have extraneous solutions because the
denominator of a fraction cannot be zero.
--------------------------------------------------------------------------------------------------------------------4 10
Solve  .
x 33
Step 1 does not need to be completed - each numerator/denominator only has one term.
4 10

x 33
2)Cross multiply.
4  33  x 10
Simplify after cross multiplying.
132  10x
3A)Since this equation is linear, isolate x thru division.
132 10 x

10
10
x  13.2
4 10

x 33
4)Check your answer by substituting 13.2 into the original
equation for x.
4 ??? 10
13.2  33
A calculator comes in handy at this point.
Therefore, x  13.2 is a solution.
.30  .30
--------------------------------------------------------------------------------------------------------------------10 5
Solve
.

7
2y
--------------------------------------------------------------------------------------------------------------------a 3 5
 .
Solve
Place parentheses around quantities of more than one term.
2
8
(a  3) 5

2
8
Cross multiply and simplify.
(a  3)  8  2  5
8a  24  10
Isolate a since this equation is linear.
8a  34
Check your solution. Use Step 1 for calculator entry.
a  4.25
(4.25  3) ??? 5
8
2
.625 = .625
Thus, a  4.25 is a solution.
--------------------------------------------------------------------------------------------------------------------8 5b  1
4
5

Solve 
.
Solve
.
5
10
3c 4c  7
Notes on Solving Rational Equations - Page 2
Solve
d 2 3
 .
5
d
( d  2) 3

5
d
Name_________________________
Place parentheses around parts with two or more terms.
Cross multiply and simplify.
(d  2)  d  5  3
d 2  2d  15
Since this equation is a quadratic equation, collect all the
terms on one side and solve.
d 2  2d  15  0
(d  5)(d  3)  0
d  5 and d  3
Check the two solutions. Use step 1 for calculator entry.
--------------------------------------------------------------------------------------------------------------------m 16
n3
10
 .

Solve
Solve
4 m
2
n2
--------------------------------------------------------------------------------------------------------------------x5 2
 , and he/she gets x = 0 as one of the
Suppose one is solving the following equation:
3x
x
possible solutions.
However, if "0" is substituted into the original equation for x, the denominator of the second
2
fraction, , will end up equaling zero. This cannot occur (you cannot divide by zero).
x
Thus, x = 0 is not a solution to this equation. This is one reason why it is important to check
every answer to every problem.
--------------------------------------------------------------------------------------------------------------------Homework on Solving Rational Equations
Solve for the variable.
1.
11 16

x 45
2.
3 y 102

2
5
4 
4.
3
 20
4a
5.
15
6

b  1 13
6.
4
14

7 c5
7.
3d  9 18

2
23
8.
2m  3 4

m
5
9.
4n  1 n  6

14
18
10.
p
p

p  10 99
11.
14
q

q3 2
12.
4r r 2  12

3
r
13.
3 5s

2 s 8s
14.
t 5
7

3
t 1
15.
3u
u2

2 u 1
16.
1
x  10

x4
x
3.
9x
2
Hint: make -4 into a fraction.
8
9
243 81

7. d 
69 23
4. a  0.0375
10. p  0; p  89
11. q  7; q  4
12. r  6; r  6
14. t  2; t  8
15. u  0; u  3
16. x  5; x  8
1. x  30.9375
2. y  13.6
5. b  31.5
6. c  29.5
102 51

58 29
13. s  2.4
9. n 
3. x  
8. m  2.5