1. 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

2. 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

3. 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 ≠ a ≠ ± x ≠ 0
6.4

4. Clearing Factions from Equations
Review: Simplify - Clear a Complex Fraction by
Multiplying top and bottom by the LCD
Solve - Clear a Rational Equation by
Multiplying both sides by the LCD
Then solve the new polynomial equation using the principle of zero products
6.4

5. 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

6. Clearing & Solving a Rational Equation
What gets excluded?
x ≠ 0
What’s the LCD?
15x
What’s the solution?
6.4

7. A Binomial Denominator
What gets excluded?
x ≠ 5
What’s the LCD?
x – 5
What’s the solution?
6.4

8. Another Binomial Denominator
What gets excluded?
x ≠ 3
What’s the LCD?
x – 3
What’s the solution?
x = -3 (x = 3 excluded)
6.4

9. Different Binomial Denominators
What gets excluded?
x ≠ 5,-5
What’s the LCD?
(x – 5)(x + 5)
What’s the solution?
x = 7
6.4

10. Functions as Rational Equations
What gets excluded?
x ≠ 0
What’s the LCD? x
What’s the solution?
x = 2 and x = 3
6.4

11. What Next?
6.5 Solving Applications of Rational Equations
6.4