1. October 31 st copyright2009merrydavidson

2. Simplifying Rational Expressions What is the difference between a factor and a term? TERMS are separated by +, - signs. FACTORS are separated by multiplication sign. An expression has NO equal sign.

3. The REDUCTION PROPERTY OF FRACTIONS You are allowed to divide out (cancel) common factors (not common terms) that appear in both the numerator and the denominator. You may NOT cancel the “4x”’s. They are terms not factors!

4. Example 2: Can you cancel anything out? Why or Why not??? There are NO add or subtract signs so these are all factors. Reduce the 2/4 to ½ and cancel out the p’s. List domain restrictions. domain restrictions are called critical numbers.

5. CRITICAL NUMBERS: Found in the numerator by solving for x. C.N. = 3/2 Found in the denominator by solving for x. C.N. = ¾,-5

6. Example 3: SIMPLIFY: List domain restrictions. List critical numbers. DO domain restrictions and critical numbers before canceling.

7. Example 4: Can you cancel anything out? Why or Why not? These are all terms so no. Factor to see if anything will match to cancel. List domain restrictions. List critical numbers. SIMPLIFY:

8. Example 5: For this example only simplify and list answer with domain restrictions. Remember to do domain restrictions BEFORE canceling!!!

9. Example 6: (x – 5) and (5 – x) are opposites….. In order to cancel you must factor out a (-1) from one of those terms.

10. 7. Solve the following rational equation. Step 1: Find the LCD. Step 2: Multiply each term by the LCD over 1. Step 3: Cancel Step 5: Solve the equation. LCD = 6 3 2 Step 4: Re-write the equation. 3x + 4 = 42 x = 38/3

11. 8. Solve the following rational equation. LCD is: 63 2 3(x + 1) + 4 = 6x 3x + 3 + 4 = 6x 7 = 3x 7/3 = x

12. Extraneous Solutions: You are not allowed to have a zero in the denominator of a fraction. Therefore, If you get x = 5 and 5 would make the denominator = 0, 5 would be an extraneous solution. In other words, if algebraically you get a solution, but that makes the denominator zero it is called an extraneous solution. For any fraction with a variable in the denominator you must list domain restrictions first (that is what x can not be).

13. 9. Solve the following rational equation. There is a variable in the denominator. What can x not be? x = 0 LCD is 5x 15 + 8x = 65 By using the LCD and canceling you eliminate the denominator. 8x = 50 x = 25/4 Since the answer is not a domain restriction we are done.

14. 10. Solve the following rational equation. List domain restrictions FIRST. Solve Check for extraneous solutions. x – 5 + 5 = 2x - 5 x = 2x - 5 5 = x But x cannot be 5. the answer is: no real solution and 5 is extraneous.

15. 11. Solve the following rational equation. Factor everything possible. domain restrictions: x = 0, -2, -1

16. LCD: (x + 1)(x + 1) – (x + 2)(x + 4) = -3x x 2 + 2x + 1 – x 2 – 6x - 8 = -3x x = -7 domain restrictions: x = 0, -2, -1

17. 12. Solve the following rational equation. Be sure that everything in the denominator gets canceled out

18. 12. 3x 2 – 13x + 4 – 3x – 3 = 2x 2 + x - 15 (x – 4)(3x – 1) – 3(x + 1) = (x + 3)(2x – 5) x 2 – 17x + 16 = 0 x = 1, 16

19. HW: WS 4-4