1. Lesson 4-6 Rational Equations and Partial Fractions
Objective: To solve rational equations and inequalities
To decompose a fraction into partial fractions

2. Rational Equations
Rational Equation – has 1 or more rational expressions.
Solve by multiplying each side by the LCD

3. Rational Equations To solve a rational equation:
1. Find the LCM of the denominators.
2. Clear denominators by multiplying both sides of the equation by the LCM.
3. Solve the resulting polynomial equation.
4. Check the solutions.

4. Find the LCM. Multiply by LCM = (x – 3). Solve for x. Check.
Examples: 1. Solve:
LCM = x – 3.
Find the LCM.
1 = x + 1
Multiply by LCM = (x – 3).
x = 0
Solve for x.
(0)
Check.
Substitute 0.
Simplify.
True.
2. Solve:
LCM = x(x – 1).
Find the LCM.
Multiply by LCM.
x – 1 = 2x
Simplify.
x = –1
Solve.
Examples: Solve

5. Check. x = 3 is not a solution since both sides would be undefined.
Example: Solve:
x2 – 8x + 15 = (x – 3)(x – 5)
Factor.
The LCM is (x – 3)(x – 5).
Original Equation.
x(x – 5) = – 6
Polynomial Equation.
Simplify.
x2 – 5x + 6 = 0
Factor.
(x – 2)(x – 3) = 0
Check. x = 2 is a solution.
x = 2 or x = 3
Check. x = 3 is not a solution since both sides would be undefined.
Example: Solve

6. Decomposing a fraction into Partial Fractions.
Sometimes we need more tools to help with rational expressions…
We will learn to perform a process known as partial fraction decomposition…
To find partial fractions for an expression, we need to reverse the process of adding fractions.

7. To find the partial fractions, we start with
The expressions are equal for all values of x so we have an identity.
The identity will be important for finding the values of A and B.

8. To find the partial fractions, we start with
Multiply by the LCD
So,
If we understand the cancelling, we can in future go straight to this line from the 1st line.

9. This is where the identity is important.
The expressions are equal for all values of x, so I can choose to let x = 2.
Why should I choose x = 2 ?
ANS: x = 2 means the coefficient of B is zero, so B disappears and we can solve for A.

10. This is where the identity is important.
The expressions are equal for all values of x, so I can choose to let x = 2.
What value would you substitute next ?
ANS: x = - 1 so that the first term becomes 0.

11. This is where the identity is important.
The expressions are equal for all values of x, so I can choose to let x = 2.
So,

12. Example 2 Express the following as 2 partial fractions.
Solution: Let
Multiply by :

13. Decomposing Fractions
Decompose into partial fractions
2/(x+2) +4/(x-5)

14. Rational Inequalities
To solve rational inequalities:
Find the zeros and mark on a number line.
Find any exclusions (restrictions) and mark on a number line.
Test a value on each interval.

15. Rational Inequalities
Solve
Set to LCD=15b
Find the zeros

16. Rational Inequalities
-1/15
Test b<-1/15
try b=-1
True
Test -1/15<b<0
try b=-1/30
False
Test b>0
try b=1
True

17. Rational InequalitiesSo

18. Rational Inequalities
Solve
-2<x<-1, -1<x<1,1<x<3, x>3