Solving Rational Equations
Remark 1 This module is devoted to solving equations! We will learn how to solve
two new types of equations: rational equations and radical equations. Then there will be a mixed review where we solve all different
types of equations (including linear and quadratic equations) in the
same topic! Finally there will be an applications topic.
Definition A rational equation is any equation that has a rational expression (a fraction with variables).
ex.
−→
1
1
+
= 12,
x 5x
x
2
1
+
=
x2 − x x2 − 1 x3 − x2
Remark 2 Actually, we’ve dealt with very simple rational equations before!
x x
ex.
−→ Solve the equation. + = 4
3 2
This is both a linear equation (because the exponent on the x
is 1), and a rational equation (because there are rational expressions in the equation).
If you recall, to solve such an equation, we cleared the denominators
by multiplying both sides of the equation by the LCD.
multiply
2x
3x
x x
x
x
by LCD
reduce
+ = 10 −→ (6)· +(6)· = (6)·10 −→ (6)·
+(
6)·
= (6)·10
3 2
3
2
3
2
multiply
combine
like terms
−→ 2x + 3x = 60 −→
divide
5x = 60 −→
x = 12
This same strategy works for solving more complicated rational expressions!
Main Idea To solve rational equations: multiply both sides of the
equation by the LCD to clear the denominators; then solve the
resulting equation.
• Remember, when finding the LCD all polynomial denominators must be factored!
• Beware of extraneous solutions (see remark 4).
Remark 3 Let’s look at a quick example.
ex.
−→
Solve the equation.
5
3
1
+
=
m2 4m 8m
The LCD for rational expressions in the equation is 8m2 . So, we mul2
tiply both sides by 8m2 (which you may think of as 8m1 if it helps).
5
3
1
+
=
m2 4m 8m
reduce
multiply
by LCD
−→
(8m2 )·
5
3
1
2
2
+
(8m
)·
=
(8m
)·
m2
4m
8m
*8
2m 3
m 1
5
*
*
2
2
+
(
8m
)
·
=
(
8m
)
· m2
4m
8m
2 )·
−→ (
8m
multiply
−→ 40 + 6m = m
get variables
on one side
−→
divide
40 = −5m −→
m = −8
Remark 4 An extraneous solution to an equation is a possible solution that
does not actually satisfy the original equation.
ex.
−→
Solve the equation.
x
4
−5=
x−4
x−4
Notice first that x − 4 is the only denominator so it is the LCD.
x
4
−5=
x−4
x−4
multiply
multiply
by LCD
−→
−→ x − 5x + 20 = 4
1 x
:
(
x
−4)·
get variables
on one side
−→
x
−
4
1 4
:
− 5·(x − 4) = (
x
−4)·
x
−
4
divide
−4x = −16 −→ x = 4
x = 4 is an extraneous solution to this equation because substituting
4 into the original equation for x gives:
4
4
4
4
simplify
−5=
−→
−5= .
4−4
4−4
0
0
4
0
makes no sense and therefore the entire equation becomes a false
statement! This means the equation has no solution .
→→→→ When solving rational equations: always exclude any soluImportant tions that make any denominator in the original equation
→→→→
zero!
Remark 5 It is important to recognize the difference between rational expressions and rational equations.
Rational expression:
ex.
−→
5
6x
+
Rational equation:
4
5
3x
6x
+
4
3x
= 10 .
• Rational expressions can be simplified. When simplifying rational expressions, the result is always another rational expression!
• Rational equations can be solved! When solving a rational equation, the result is a set of x-values.
→→→→ When working with rational expressions (not equations), the
Important
→→→→ denominators CANNOT be cleared!
Example 1 Identify each of the following as an expression or an equation.
Then simplify the expression or solve the equation.
a
2x 9x
−
= 19
3
4
b
2x 9x
−
3
4
Example 2 Solve each equation.
solutions.
Be sure to exclude any extraneous
a
2
3
5
+ =
p + 3 8 4p + 12
b
4
13
3
− 2
=
t + 2 t + 6t + 8 2t + 8
c
x
4
18
+
= 2
x−3 x+3 x −9