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DO NOW 02/29/2016 Add or subtract.

Learn how to add and subtract rational expressions, simplify complex fractions, find common denominators, and identify undefined values. Examples and step-by-step instructions included.

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DO NOW 02/29/2016 Add or subtract.

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  1. + 7 11 2 1 3 13 13 – 12 15 5 3 8 15 24 x– 1 4x9 1 x6 x2 – 1 x + 1 12x3 DO NOW 02/29/2016 Add or subtract. 1. 2. Simplify. Identify any x-values for which the expression is undefined. 3. x ≠ 0 4. x ≠ –1, x ≠ 1

  2. Objectives Add and subtract rational expressions. Simplify complex fractions.

  3. Adding and subtracting rational expressions are similar to adding and subtracting fractions. With like denominators, add or subtract the numerators and use the same denominator.

  4. + x2 – 3 6x + 5 + 6x + 5 3x – 1 3x – 1 9x + 4 x2 – 3 x2 – 3 x2 – 3 Example 1a Add or subtract. Add the numerators. Combine like terms.

  5. – 3x – 1 (2x2– 3x – 2) 2x2 +3x + 2 2x2–3x –2 x2 + 3x – 3 3x2– 5 3x2– 5 – 3x2– 5 – 3x – 1 3x – 1 3x – 1 3x – 1 Example 1b Add or subtract. Subtract the numerators. Distribute the negative sign. Combine like terms.

  6. To add or subtract rational expressions with unlike denominators, first find the least common denominator. The LCD is the least common multiple of the polynomials in the denominators.

  7. Example 2 Find the least common multiple for each pair. a. 4x3y7 and 3x5y4 4x3y7 = 2 2 x3  y7 3x5y4 = 3 x5  y4 The LCM is 2 23 x5  y7, or 12x5y7. b. x2 – 4 and x2 + 5x + 6 x2 – 4 = (x – 2)(x + 2) x2 + 5x + 6= (x + 2)(x + 3) The LCM is (x – 2)(x + 2)(x + 3).

  8. + + + The LCD is 6(x – 1), so multiply by 3 and by 2. 3x – 2 3x – 2 3x – 2 3x 3x 3x 2 3 3x – 2 3x 3(x – 1) 3(x – 1) 2(x – 1) 2(x – 1) 3x – 3 2x –2 2 3 3(x – 1) 2(x – 1) Example 3a Add. Factor the denominators.

  9. 9x + 6x – 4 15x – 4 6(x – 1) 6(x – 1) Example 3a Continued Add the numerators. Simplify the numerator.

  10. + + x2+ 3x + 2x + 6 2x + 6 2x + 6 2x + 6 x + 3 x x x (x + 3) x The LCD is (x + 3)(x + 3), so multiply by . (x + 3)(x + 3) (x+ 3)(x + 3) (x+ 3)(x + 3) x2 + 6x + 9 x + 3 x + 3 x + 3 x + 3 (x + 3) (x + 3) + Example 3b Add. Identify any x-values for which the expression is undefined. Factor the denominators. Add the numerators.

  11. (x + 3)(x + 2) x2+ 5x + 6 x + 2 (x + 3)(x + 3) (x+ 3)(x + 3) x + 3 Example 3b Continued Add. Identify any x-values for which the expression is undefined. Write the numerator in standard form. Factor the numerator. Divide out common factors. The expression is undefined at x = –3 because this value of x make the factors (x + 3) and (x + 3) equal 0.

  12. Subtract 2x2 + 64 2x2 + 64 2x2 + 64 x – 4 x –4 x – 4 – – – (x–8)(x +8) (x–8)(x +8) x2 –64 x + 8 x + 8 x + 8 x – 8 (x – 8) x – 4 2x2 + 64 – (x2 – 12x + 32) 2x2 + 64 – (x – 4)(x – 8) The LCD is (x – 3)(x + 8), so multiply by . x – 8 (x – 8) x + 8 (x –8)(x +8) (x –8)(x +8) Example 4b Factor the denominators. Subtract the numerators. Multiply the binomials in the numerator.

  13. Subtract 2x2 + 64 x –4 – x2 –64 x + 8 x + 4 2x2 + 64 –x2 + 12x – 32 (x + 8)(x + 4) x2 + 12x + 32 x – 8 (x –8)(x +8) (x –8)(x +8) (x –8)(x + 8) Distribute the negative sign. Write the numerator in standard form. Factor the numerator. Divide out common factors. The expression is undefined at x = 8 and x = –8 because these values of x make the factors (x + 8) and (x – 8) equal 0.

  14. Some rational expressions are complex fractions. A complex fraction contains one or more fractions in its numerator, its denominator, or both. Examples of complex fractions are shown below. Recall that the bar in a fraction represents division. Therefore, you can rewrite a complex fraction as a division problem and then simplify. You can also simplify complex fractions by using the LCD of the fractions in the numerator and denominator.

  15. x + 1 x x2 – 1 x – 1 ÷  x + 1 x + 1 x – 1 x x2 – 1 x2 – 1 x – 1 x Example 5a Simplify. Assume that all expressions are defined. Write the complex fraction as division. Write as division. Multiply by the reciprocal.

  16. x + 1 x – 1 1 (x – 1)(x + 1) x x  Example 5a Continued Simplify. Assume that all expressions are defined. Factor the denominator. Divide out common factors.

  17. 20 6 3x – 3 x – 1 ÷  3x – 3 20 20 6 3x – 3 x – 1 x – 1 6 Example 5b Simplify. Assume that all expressions are defined. Write the complex fraction as division. Write as division. Multiply by the reciprocal.

  18. 3(x – 1) 20 x – 1 6 Example 5b Continued Simplify. Assume that all expressions are defined. Factor the numerator. Divide out common factors. 10

  19. 1 1 1 1 + x + 4 x + 4 2x 2x x x x – 2 x – 2 + (2x)(x – 2) (2x)(x – 2) (2x)(x – 2) Check It Out! Example 5c Simplify. Assume that all expressions are defined. Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator. The LCD is (2x)(x – 2).

  20. (2)(x – 2) + (x – 2) 3(x – 2) 3x – 6 (x + 4)(2x) (x + 4)(2x) 2x(x + 4) Check It Out! Example 5c Continued Simplify. Assume that all expressions are defined. Divide out common factors. or Simplify.

  21. d d d d 45 45 40 40 Total time: Use the formula t = . + + d total distance r total time 2d Average speed: The average speed is . Example 6 Justin’s average speed on his way to school is 40 mi/h, and his average speed on the way home is 45 mi/h. What is Justin’s average speed for the entire trip? Round to the nearest tenth. Total distance: 2d Let d represent the one-way distance.

  22. 2d(360) (360) + (360) d d 45 40 720d 720d 9d + 8d 17d ≈ 42.4 Example 6 The LCD of the fractions in the denominator is 360. Simplify. Combine like terms and divide out common factors. Justin’s average speed is 42.4 mi/h.

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