Adding and Subtracting 8 3 Rational Expressions Warm

Adding and Subtracting 8 -3 Rational Expressions Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

8 -3 Adding and Subtracting Rational Expressions Warm Up Add or subtract. 2 + 5 11 – 2. 12 1. 7 15 3 8 13 15 13 24 Simplify. Identify any x-values for which the expression is undefined. 9 4 x 3. 1 x 6 x ≠ 0 3 12 x 3 4. x– 1 x 2 – 1 Holt Mc. Dougal Algebra 2 1 x+1 x ≠ – 1, x ≠ 1

8 -3 Adding and Subtracting Rational Expressions Objectives Add and subtract rational expressions. Simplify complex fractions. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Vocabulary complex fraction Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 1 A: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. x– 3 + x– 2 x+4 x– 3+ x– 2 Add the numerators. x+4 2 x – 5 x+4 Combine like terms. The expression is undefined at x = – 4 because this value makes x + 4 equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 1 B: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. 3 x – 4 – 6 x + 1 x 2 + 1 3 x – 4 – (6 x + 1) x 2 + 1 3 x – 4 – 6 x – 1 x 2 + 1 – 3 x – 5 x 2 + 1 There is no real value of the expression is always Holt Mc. Dougal Algebra 2 Subtract the numerators. Distribute the negative sign. Combine like terms. x for which x 2 + 1 = 0; defined.

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 1 a Add or subtract. Identify any x-values for which the expression is undefined. 6 x + 5 + 3 x – 1 x 2 – 3 Add the numerators. 9 x + 4 x 2 – 3 Combine like terms. The expression is undefined at x = ± this value makes x 2 – 3 equal 0. Holt Mc. Dougal Algebra 2 because

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 1 b Add or subtract. Identify any x-values for which the expression is undefined. 3 x 2 – 5 – 2 x 2 – 3 x – 2 3 x – 1 3 x 2 – 5 – (2 x 2 – 3 x – 2) 3 x – 1 3 x 2 – 5 – 2 x 2 + 3 x + 2 3 x – 1 x 2 + 3 x – 3 3 x – 1 Subtract the numerators. Distribute the negative sign. Combine like terms. The expression is undefined at x = 1 because 3 this value makes 3 x – 1 equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 2: Finding the Least Common Multiple of Polynomials Find the least common multiple for each pair. A. 4 x 2 y 3 and 6 x 4 y 5 4 x 2 y 3 = 2 2 x 2 y 3 6 x 4 y 5 = 3 2 x 4 y 5 The LCM is 2 2 3 x 4 y 5, or 12 x 4 y 5. B. x 2 – 2 x – 3 and x 2 – x – 6 x 2 – 2 x – 3 = (x – 3)(x + 1) x 2 – x – 6 = (x – 3)(x + 2) The LCM is (x – 3)(x + 1)(x + 2). Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 2 Find the least common multiple for each pair. a. 4 x 3 y 7 and 3 x 5 y 4 4 x 3 y 7 = 2 2 x 3 y 7 3 x 5 y 4 = 3 x 5 y 4 The LCM is 2 2 3 x 5 y 7, or 12 x 5 y 7. b. x 2 – 4 and x 2 + 5 x + 6 x 2 – 4 = (x – 2)(x + 2) x 2 + 5 x + 6 = (x + 2)(x + 3) The LCM is (x – 2)(x + 3). Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 3 A: Adding Rational Expressions Add. Identify any x-values for which the expression is undefined. x– 3 2 x + x 2 + 3 x – 4 x+4 x– 3 2 x + (x + 4)(x – 1) x + 4 Holt Mc. Dougal Algebra 2 Factor the denominators. x – 1 The LCD is (x + 4)(x – x 1), 2 x by – 1. so multiply x– 1 x+4

8 -3 Adding and Subtracting Rational Expressions Example 3 A Continued Add. Identify any x-values for which the expression is undefined. x – 3 + 2 x(x – 1) (x + 4)(x – 1) Add the numerators. 2 x 2 – x – 3 (x + 4)(x – 1) Simplify the numerator. Write the sum in factored or expanded form. The expression is undefined at x = – 4 and x = 1 because these values of x make the factors (x + 4) and (x – 1) equal 0. 2 x 2 – x – 3 or (x + 4)(x – 1) Holt Mc. Dougal Algebra 2 2 x 2 – x – 3 x 2 + 3 x – 4

8 -3 Adding and Subtracting Rational Expressions Example 3 B: Adding Rational Expressions Add. Identify any x-values for which the expression is undefined. x + 2– 8 x+2 x – 4 x + – 8 x + 2 (x + 2)(x – 2) Factor the denominator. x – 8 x– 2 + The LCD is (x + 2)(x – 2), x x+ 2 x – 2 (x + 2)(x – 2) so multiply by x – 2. x– 2 x+2 x(x – 2) + (– 8) (x + 2)(x – 2) Holt Mc. Dougal Algebra 2 Add the numerators.

8 -3 Adding and Subtracting Rational Expressions Example 3 B Continued Add. Identify any x-values for which the expression is undefined. x 2 – 2 x – 8 (x + 2)(x – 2) Write the numerator in standard form. (x + 2)(x – 4) (x + 2)(x – 2) Factor the numerator. x– 4 x– 2 Divide out common factors. The expression is undefined at x = – 2 and x = 2 because these values of x make the factors (x + 2) and (x – 2) equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 3 a Add. Identify any x-values for which the expression is undefined. 3 x + 3 x – 2 2 x – 2 3 x – 3 3 x + 3 x – 2 2(x – 1) 3 x 3 + 3 x – 2 2 2(x – 1) 3 3(x – 1) 2 Holt Mc. Dougal Algebra 2 Factor the denominators. The LCD is 6(x – 1), so 3 x multiply by 3 and 2(x – 1) 3 x – 2 by 2. 3(x – 1)

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 3 a Continued Add. Identify any x-values for which the expression is undefined. 9 x + 6 x – 4 6(x – 1) 15 x – 4 6(x – 1) Add the numerators. Simplify the numerator. The expression is undefined at x = 1 because this value of x make the factor (x – 1) equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 3 b Add. Identify any x-values for which the expression is undefined. x x+3 + 2 x + 6 x 2 + 6 x + 9 x x+3 2 x + 6 + (x + 3) Factor the denominators. x 2 x + 6 x+3+ The LCD is (x + 3), x + 3 (x + 3) x (x + 3) so multiply (x + 3) by (x + 3). x 2 + 3 x + 2 x + 6 (x + 3) Holt Mc. Dougal Algebra 2 Add the numerators.

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 3 b Continued Add. Identify any x-values for which the expression is undefined. x 2 + 5 x + 6 (x + 3)(x + 2) (x + 3) x+2 x+3 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. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 4: Subtracting Rational Expressions 2 – 30 2 x x + 5. Identify any x. Subtract – 2 x – 9 x+3 values for which the expression is undefined. 2 x 2 – 30 x+5 – Factor the denominators. x+3 (x – 3)(x + 3) 2 x 2 – 30 x + 5 x – 3 The LCD is (x – 3)(x + 3), – x+5 (x – 3)(x + 3) x + 3 x – 3 so multiply x + 3 by (x – 3). 2 x 2 – 30 – (x + 5)(x – 3)(x + 3) Subtract the numerators. 2 x 2 – 30 – (x 2 + 2 x – 15) (x – 3)(x + 3) Multiply the binomials in the numerator. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 4 Continued 2 – 30 x + 5. Identify any x 2 x Subtract – x 2 – 9 x+3 values for which the expression is undefined. 2 x 2 – 30 – x 2 – 2 x + 15 Distribute the negative sign. (x – 3)(x + 3) x 2 – 2 x – 15 Write the numerator in (x – 3)(x + 3) standard form. (x + 3)(x – 5) Factor the numerator. (x – 3)(x + 3) x– 5 Divide out common factors. x– 3 The expression is undefined at x = 3 and x = – 3 because these values of x make the factors (x + 3) and (x – 3) equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 4 a 3 x – 2 2. Identify any x– 2 x + 5 5 x – 2 values for which the expression is undefined. Subtract 3 x – 2 5 x – 2 2 2 x + 5 The LCD is (2 x + 5)(5 x – 2), – 2 x + 5 5 x – 2 2 x + 5 so multiply 3 x – 2 by (5 x – 2) 2 x + 5 (5 x – 2) 2 by (2 x + 5). and 5 x – 2 (2 x + 5) (3 x – 2)(5 x – 2) – 2(2 x + 5)(5 x – 2) Subtract the numerators. 15 x 2 – 16 x + 4 – (4 x + 10) (2 x + 5)(5 x – 2) Multiply the binomials in the numerator. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 4 a Continued 3 x – 2 2. Identify any x– 2 x + 5 5 x – 2 values for which the expression is undefined. Subtract 15 x 2 – 16 x + 4 – 4 x – 10 (2 x + 5)(5 x – 2) Distribute the negative sign. The expression is undefined at x = – 5 and x = 2 2 5 because these values of x make the factors (2 x + 5) and (5 x – 2) equal 0. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 4 b 2 + 64 x – 4. Identify any x 2 x Subtract – 2 x – 64 x+8 values for which the expression is undefined. 2 x 2 + 64 x– 4 – (x – 8)(x + 8) x+8 Factor the denominators. The LCD is (x – 3)(x + 8), x– 8 x– 4 (x – 8) x – 8 so multiply x + 8 by (x – 8). 2 x 2 + 64 – (x – 4)(x – 8)(x + 8) Subtract the numerators. 2 x 2 + 64 – (x 2 – 12 x + 32) (x – 8)(x + 8) Multiply the binomials in the numerator. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 4 b 2 + 64 2 x x – 4. Identify any x. Subtract – x 2 – 64 x+8 values for which the expression is undefined. 2 x 2 + 64 – x 2 + 12 x – 32) Distribute the negative sign. (x – 8)(x + 8) x 2 + 12 x + 32 Write the numerator in (x – 8)(x + 8) standard form. (x + 8)(x + 4) Factor the numerator. (x – 8)(x + 8) x+4 Divide out common factors. x– 8 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. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions 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. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Example 5 A: Simplifying Complex Fractions Simplify. Assume that all expressions are defined. x+2 x– 1 x– 3 x+5 Write the complex fraction as division. x+2 ÷ x– 3 Write as division. x– 1 x+5 Multiply by the x+2 x+5 reciprocal. x– 1 x– 3 (x + 2)(x + 5) or x 2 + 7 x + 10 (x – 1)(x – 3) x 2 – 4 x + 3 Holt Mc. Dougal Algebra 2 Multiply.

8 -3 Adding and Subtracting Rational Expressions Example 5 B: Simplifying Complex Fractions Simplify. Assume that all expressions are defined. 3 x x x + 2 – 1 x Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator. 3 x (2 x) + (2 x) x 2 x – 1 (2 x) x Holt Mc. Dougal Algebra 2 The LCD is 2 x.

8 -3 Adding and Subtracting Rational Expressions Example 5 B Continued Simplify. Assume that all expressions are defined. (3)(2) + (x)(x) (x – 1)(2) Divide out common factors. x 2 + 6 or 2(x – 1) Simplify. Holt Mc. Dougal Algebra 2 x 2 + 6 2 x – 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 a Simplify. Assume that all expressions are defined. x+1 x 2 – 1 x x– 1 Write the complex fraction as division. x+1 ÷ x x 2 – 1 x– 1 Write as division. x+1 2 x – 1 Multiply by the reciprocal. x– 1 x Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 a Continued Simplify. Assume that all expressions are defined. x+1 x– 1 (x – 1)(x + 1) x 1 x Holt Mc. Dougal Algebra 2 Factor the denominator. Divide out common factors.

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 b Simplify. Assume that all expressions are defined. 20 x– 1 6 3 x – 3 Write the complex fraction as division. 20 ÷ 6 x– 1 3 x – 3 Write as division. 20 3 x – 3 x– 1 6 Multiply by the reciprocal. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 b Continued Simplify. Assume that all expressions are defined. 3(x – 1) 20 6 x– 1 10 Holt Mc. Dougal Algebra 2 Factor the numerator. Divide out common factors.

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 c Simplify. Assume that all expressions are defined. 1 1 + x 2 x x+4 x– 2 Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator. 1 1 (2 x)(x – 2) + (2 x)(x – 2) x 2 x x + 4 (2 x)(x – 2) x– 2 Holt Mc. Dougal Algebra 2 The LCD is (2 x)(x – 2).

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 5 c Continued Simplify. Assume that all expressions are defined. (2)(x – 2) + (x – 2) (x + 4)(2 x) 3 x – 6 or 3(x – 2) (x + 4)(2 x) 2 x(x + 4) Holt Mc. Dougal Algebra 2 Divide out common factors. Simplify.

8 -3 Adding and Subtracting Rational Expressions Example 6: Transportation Application A hiker averages 1. 4 mi/h when walking downhill on a mountain trail and 0. 8 mi/h on the return trip when walking uphill. What is the hiker’s average speed for the entire trip? Round to the nearest tenth. Total distance: 2 d Let d represent the one-way distance. Total time: d + d 1. 4 0. 8 Use the formula t = d r. Average speed: Holt Mc. Dougal Algebra 2 2 d d + d 1. 4 0. 8 The average speed is total distance. total time

8 -3 Adding and Subtracting Rational Expressions Example 6 Continued 2 d d = 5 d and d = 5 d. 7 4 1. 4 7 0. 8 4 5 5 2 d(28) The LCD of the fractions in the 5 d (28) + 5 d (28) denominator is 28. 7 4 d + d 1. 4 0. 8 56 d 20 d + 35 d 55 d ≈ 1. 0 55 d Simplify. Combine like terms and divide out common factors. The hiker’s average speed is 1. 0 mi/h. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Check It Out! 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: 2 d Let d represent the one-way distance. Total time: d + d 40 45 Use the formula t = d r. Average speed: Holt Mc. Dougal Algebra 2 2 d d + d 40 45 The average speed is total distance. total time

8 -3 Adding and Subtracting Rational Expressions Check It Out! Example 6 2 d(360) d (360)+ d (360) 40 45 720 d 9 d + 8 d 720 d ≈ 42. 4 17 d 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. Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Lesson Quiz: Part I Add or subtract. Identify any x-values for which the expression is undefined. 1. 2 x + 1 + x – 3 x– 2 x+1 2. x x+4 – x 2 + 36 x – 16 3 x 2 – 2 x + 7 x ≠ – 1, 2 (x – 2)(x + 1) x– 9 x– 4 x ≠ 4, – 4 3. Find the least common multiple of x 2 – 6 x + 5 and x 2 + x – 2. (x – 5)(x – 1)(x + 2) Holt Mc. Dougal Algebra 2

8 -3 Adding and Subtracting Rational Expressions Lesson Quiz: Part II 4. Simplify defined. x+2 x 2 – 4 x x– 2 . Assume that all expressions are 1 x 5. Tyra averages 40 mi/h driving to the airport during rush hour and 60 mi/h on the return trip late at night. What is Tyra’s average speed for the entire trip? 48 mi/h Holt Mc. Dougal Algebra 2