1. Quiz 8-5 1. 2. Simplify 3.

2. 8-6 Solve Rational Equations

3. Vocabulary: What does solve a single variable equation mean? What is a factor? What is a least common multiple? 3x + 2 = 11

4. Solving Rational Equations Method #1: eliminate the denominators one at a time. Method #2: Obtain common denominators for each term. Method #3: Find the least common multiple of all the denominators then multiply (left/right) by the LCM. denominators then multiply (left/right) by the LCM.

5. Rational Equations How do you get the ‘5’ out of the denominator? Multiply both sides by ‘5’ x = 20

6. Rational Equations How do you get the ‘x’ out of the denominator? Multiply both sides by ‘x’ 2x = 10 x = ?

7. Rational Equations How do you get the ‘x-2’ out of the denominator? Multiply both sides by ‘x-2’ 4(x – 2) = 12 ÷4 ÷4 x – 2 = 3 +2 +2 x = 5

8. Your turn: 1. Solve 2. Solve 4. Solve 3. Solve

9. Variable on both sides of the equation How do you get the ‘x’ out of the left-side denominator? left-side denominator? Multiply both sides by ‘x’

10. Variable on both sides of the equation How do you get the (x+1) out of the right-side denominator? right-side denominator? Multiply both sides by (x+1)

11. Another example: The book teaches you to “cross-multiply” (yuck!) There is no property called “cross multiply”. There are the properties of equality. Cross multiplication is the result of multiplying both sides by the left denominator, then multiplying both sides by the left denominator, then multiplying the both sides by the right denominator. the both sides by the right denominator. DON’T THINK Cross multiply!! Forget it completely!!

12. Eliminate one Denomiator at a time. (4x + 5) * * (4x + 5) (x + 1) * * (x + 1) Distributive property -9x both sides -15 both sides ÷3 both sides

13. Or: obtain a common denominator (4x + 5) * (4x + 5) * (4x + 5) * * (x + 1) Multiply both sides by the common denominator

14. Your Turn: 6. Solve 7. Solve 8. Solve

15. What about this wrinkle? “multiply to eliminate the denominators one at a time” x * * x Now what? It’s a quadratic, solve the quadratic! solve the quadratic! put into standard form !!! put into standard form !!! Your turn: 9. solve

16. Rational equations with 2 solutions. Eliminate denominators one at at time. Multiply left/right by (x – 5) Carefull: distributive property

17. Rational equations with 2 solutions. Eliminate denominators one at at time. Multiply left/right by (x – 5) Carefull: distributive property

18. Rational equations with 2 solutions. Eliminate denominators one at at time. Combine like terms. Multiply left/right x

19. Rational equations with 2 solutions. Eliminate denominators one at at time. Distributive property. What kind of equation is this? Quadratic: get into standard form then solve solve

20. Your turn: 10. Solve Put all factors inside parentheses to avoid silly distributive property errors later on.

21. Vocabulary Extranious Solution: a solution obtained algebraically but which doesn’t work in the original equation.

22. Check for Extraneous solutions. 11. Solve. 11. Solve. 12. Check to see if either one is extraneous. 12. Check to see if either one is extraneous. Your Turn:

23. Multiply for the LCD

24. Multiply both sides by the LCD!! 2. What does the 1 st denominator need to be common with the other two denominators? Needs a ‘2’ What is the least common denominator? 1. Look for what is common to each denominator. Factor the denominator denominator 3. What does the 2 nd denominator need to be common with the other two denominators? 3 Needs an ‘x’ 4. What does the 3 rd denominator need to be common with the other two denominators? Needs an ‘2’

25. Find the least common denominator 3x 2*3 3x 2 * * x The least common denominator is: The least common denominator is: 6x Multiply both sides by the LCD. 6x

26. Find the least common denominator The LCD is: The LCD is: 6x Multiply both sides by the LCD. 6x Notice how I factored 6x so that it is easier to simplify. so that it is easier to simplify. Wow ! x = 4