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Lesson 4-6 Rational Equations and Partial Fractions

Lesson 4-6 Rational Equations and Partial Fractions. Objective: To solve rational equations and inequalities To decompose a fraction into partial fractions. Rational Equations. Rational Equation – has 1 or more rational expressions. Solve by multiplying each side by the LCD.

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

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

  5. Example: Solve: . 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.

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

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

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

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

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

  13. Decomposing Fractions • Decompose into partial fractions

  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 0 LCD=15b • Find the zeros

  16. Rational Inequalities -1/15 0 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

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