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Lesson 4-6 Rational Equations and Partial Fractions

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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 Inequalities
So

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


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