1. Section 6.4 Solving Rational Expressions

2. Review) Non Permissible Values (NPV)
Any value of “x” (Variable) that makes the denominator equal to zero
Ie: Find the NPV
Look at the Denominator
Make each denominator equal to zero and solve for “x”

3. I) Solving Rational Equations
“Solving” means finding the value of “x” where both sides of an equation are equal
On a graph, it’s where the two functions cross/intersect each other
Steps to Solve Algebraically
Find the NPV (Non-Permissible Values)
Find the LCD
Multiply all terms by LCD to cancel out the Denominator
Simplify and Solve for ‘x’
‘x’ can not be the same as the NPV (Extraneous Root)

4. NOTE: Solving VS Simplifying
Simplifying: Reducing an equation to a simplified form
Solving: Finding a value for the variable so that both sides of the equation will be EQUAL!!!
Find the L.C.D.
Multiply “ALL” terms by LCD to cancel out the denominator
Check your answer by plugging it back into the equation
Find the N.P.V.

5. Ex: Solve for “x” LCD: NPV
Multiply all terms by LCD, not just terms with a denominator!
NPV

6. Practice: Solve each of the following
LCD:
LCD:

7. Challenge: Solve
LCD:

8. II) Equations with Extraneous Solutions or No Solutions
Extraneous solutions are answers that are not allowed
This happens when the solution is the same as the NPV
An equation will have no solutions when both sides are not equal when all the variables are eliminated
In contrast, an equation will have infinite solutions when both sides are equal when all variables are gone
When terms in a binomial are reversed, you get a negative in front

9. Ex: Solve and Find the NPV
No Solutions because both sides are not equal!!
Extraneous Root!!

10. Practice: Solve & Find the NPV
The terms in the binomial are switched
Both sides are exactly equal
There are no variables left and both sides are NOT equal
No matter what the value of “x”
is, both sides are always the same
Answer will be no real solutions
Solution will be all real numbers

11. Practice: Solve Find the NPV: Find the NPV: Find the LCD:
The solution is the same as the extraneous root, so, no solutions

12. III) Solving Rational Equations by Graphs
The left side of the equation is y1.
The right side of the equation is y2
Rational Function
NPV:
Straight Line x y
-30
-25
-20
-15
-10 -5 5 10 15
-12 -8 -4 4 8 12 x y
-30
-25
-20
-15
-10 -5 5 10 15
-12 -8 -4 4 8 12 x y
-30
-25
-20
-15
-10 -5 5 10 15
-12 -8 -4 4 8 12
Solving means finding the point where both functions intersect each other

13. Ex: Solve Find the NPV: Find the LCD: Multiply all terms by the LCD
Solve for “x”
The solution is not the same as the NPV

16. Homework:
Assignment 6.4