Section 6.4
Rational Equations

Solving Rational Equations





Clearing Fractions in an Equation
Restricted Domains (and Solutions)
The Principle of Zero Products
The Necessity of Checking
Rational Equations and Graphs
6.4
1
A Rational Equation in One Variable
May Have Solution(s)

A Rational Equation contains at least one
Rational Expression. Examples:

All Solution(s) must be tested in the Original Equation
6.4
2
False Solutions





Warning: Clearing an equation may add a False Solution
A False Solution is one that causes an untrue equation,
or a divide by zero situation in the original equation
Before even starting to solve a rational equation, we
need to identify values to be excluded
What values need to be excluded for these?
t≠0
a ≠ ±5
x≠0
6.4
3
Clearing Factions from Equations

Review: Simplify - Clear a Complex Fraction by


Multiplying top and bottom by the LCD
Rational Expression  LCD
Rational Expression LCD
Solve - Clear a Rational Equation by

Multiplying both sides by the LCD
LCD  Rational Expression  LCD  Rational Expression 

Then solve the new polynomial equation using the
principle of zero products
6.4
4
The Principle of Zero Products






Covered in more detail in Section 5.8
When a polynomial equation is in form
polynomial = 0
you can set each factor to zero to find solution(s)
Example – What are the solutions to:
x2 – x – 6 = 0
(x – 3)(x + 2) = 0
x – 3 = 0  x = 3 and x + 2 = 0  x = -2
6.4
5
Clearing & Solving a Rational Equation
What gets excluded?
x≠0
What’s the LCD?
15x
What’s the solution?
6.4
6
A Binomial Denominator
What gets excluded?
x≠5
What’s the LCD?
x–5
What’s the solution?
x 1
4
( x  5) 
 ( x  5) 
x 5
x 5
x 1  4
x  5 false solution
No Solution
6.4
7
Another Binomial Denominator
What gets excluded?
x≠3
What’s the LCD?
x–3
What’s the solution?
x = -3 (x = 3 excluded)
(3) 2
9

33 33
9
9

 3 Checks
6 6
2
x
9

x 3 x 3
2
x 9
x  3
6.4
8
Different Binomial Denominators
What gets excluded?
x ≠ 5,-5
What’s the LCD?
(x – 5)(x + 5)
What’s the solution?
x=7
2
1
16
Solve :

 2
x  5 x  5 x  25
2( x  5)  1( x  5)  16
2 x  10  x  5  16
2
1
16
3 x  21


7  5 7  5 49  25
2 1 16
x7
 
12 2 24
4  12 16

7 Checks
24
24
6.4
9
Functions as Rational Equations
What gets excluded?
x≠0
What’s the LCD?
x
What’s the solution?
x = 2 and x = 3
6
2
5  23
2 Checks
5 2
6
5  3
3
5  3 2
3 Checks
6
f ( x)  x 
find values when f (a )  5
x
6
5 x
x
5x  x 2  6
0  x 2  5x  6
0  ( x  2)( x  3)
x  2 and x  3
6.4
10
What Next?

6.5 Solving
Applications of
Rational Equations
6.4
11