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Adding and Subtracting Adding and Subtracting 8-3 8-3 Rational Expressions Rational Expressions Warm Up Lesson Presentation Lesson Quiz HoltMcDougal Algebra 2Algebra 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 4x 3. 1 x6 x ≠ 0 3 12x 3 4. x– 1 x2 – 1 Holt McDougal 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Vocabulary complex fraction Holt McDougal 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 1A: 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+4 x–3+x–2 Add the numerators. x+4 2x – 5 Combine like terms. x+4 The expression is undefined at x = –4 because this value makes x + 4 equal 0. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 1B: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. 3x – 4 – 6x + 1 x2 + 1 x2 + 1 3x – 4 – (6x + 1) x2 + 1 3x – 4 – 6x – 1 x2 + 1 –3x – 5 x2 + 1 There is no real value of the expression is always Holt McDougal Algebra 2 Subtract the numerators. Distribute the negative sign. Combine like terms. x for which x2 + 1 = 0; defined. 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 1a Add or subtract. Identify any x-values for which the expression is undefined. 6x + 5 + 3x – 1 x2 – 3 x2 – 3 6x + 5 + 3x – 1 x2 – 3 9x + 4 x2 – 3 Add the numerators. Combine like terms. The expression is undefined at x = ± this value makes x2 – 3 equal 0. Holt McDougal Algebra 2 because 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 1b Add or subtract. Identify any x-values for which the expression is undefined. 3x2 – 5 – 2x2 – 3x – 2 3x – 1 3x – 1 3x2 – 5 – (2x2 – 3x – 2) Subtract the numerators. 3x – 1 3x2 – 5 – 2x2 + 3x + 2 Distribute the negative sign. 3x – 1 x2 + 3x – 3 Combine like terms. 3x – 1 1 because The expression is undefined at x = 3 this value makes 3x – 1 equal 0. Holt McDougal 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 McDougal 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. 4x2y3 and 6x4y5 4x2y3 = 2 2 x2 y3 6x4y5 = 3 2 x4 y5 The LCM is 2 2 3 x4 y5, or 12x4y5. B. x2 – 2x – 3 and x2 – x – 6 x2 – 2x – 3 = (x – 3)(x + 1) x2 – x – 6 = (x – 3)(x + 2) The LCM is (x – 3)(x + 1)(x + 2). Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! 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). Holt McDougal 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 3A: Adding Rational Expressions Add. Identify any x-values for which the expression is undefined. x–3 2x + x2 + 3x – 4 x+4 x–3 2x + (x + 4)(x – 1) x + 4 x–3 2x + (x + 4)(x – 1) x + 4 Holt McDougal Algebra 2 Factor the denominators. + 4)(x – 1), x – 1 The LCD is (x2x x– 1. so multiply by x–1 x–1 x+4 8-3 Adding and Subtracting Rational Expressions Example 3A Continued Add. Identify any x-values for which the expression is undefined. x – 3 + 2x(x – 1) (x + 4)(x – 1) 2x2 – x – 3 (x + 4)(x – 1) Add the numerators. 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. 2x2 – x – 3 or (x + 4)(x – 1) Holt McDougal Algebra 2 2x2 – x – 3 x2 + 3x – 4 8-3 Adding and Subtracting Rational Expressions Example 3B: 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 McDougal Algebra 2 Add the numerators. 8-3 Adding and Subtracting Rational Expressions Example 3B Continued Add. Identify any x-values for which the expression is undefined. x2 – 2x – 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 3a Add. Identify any x-values for which the expression is undefined. 3x + 3x – 2 2x – 2 3x – 3 3x + 3x – 2 2(x – 1) 3(x – 1) 3x 3 + 3x – 2 2 2(x – 1) 3 3(x – 1) 2 Holt McDougal Algebra 2 Factor the denominators. The LCD is 6(x – 1), so multiply 3x by 3 and 2(x – 1) 3x – 2 by 2. 3(x – 1) 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 3a Continued Add. Identify any x-values for which the expression is undefined. 9x + 6x – 4 6(x – 1) 15x – 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 3b Add. Identify any x-values for which the expression is undefined. x x+3 + 2x + 6 x2 + 6x + 9 x x+3 2x + 6 + (x + 3)(x + 3) Factor the denominators. x x+3+ 2x + 6 The LCD is (x + 3)(x + 3), x + 3 x + 3 (x + 3)(x + 3) x (x + 3) so multiply (x + 3) by (x + 3) . x2 + 3x + 2x + 6 (x + 3)(x + 3) Holt McDougal Algebra 2 Add the numerators. 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 3b Continued Add. Identify any x-values for which the expression is undefined. x2 + 5x + 6 (x + 3)(x + 3) (x + 3)(x + 2) (x + 3)(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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 4: Subtracting Rational Expressions 2 – 30 2x x + 5 . Identify any xSubtract – 2 x –9 x+3 values for which the expression is undefined. 2x2 – 30 x+5 – (x – 3)(x + 3) x+3 Factor the denominators. 2x2 – 30 x + 5 x – 3 The LCD is (x – 3)(x + 3), – x+5 (x – 3) (x – 3)(x + 3) x + 3 x – 3 so multiply x + 3 by (x – 3) . 2x2 – 30 – (x + 5)(x – 3) (x – 3)(x + 3) Subtract the numerators. 2x2 – 30 – (x2 + 2x – 15) (x – 3)(x + 3) Multiply the binomials in the numerator. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 4 Continued 2 – 30 2x x + 5 . Identify any xSubtract – x2 – 9 x+3 values for which the expression is undefined. 2x2 – 30 – x2 – 2x + 15 Distribute the negative sign. (x – 3)(x + 3) x2 – 2x – 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 4a 3x – 2 2 . Identify any x– 2x + 5 5x – 2 values for which the expression is undefined. Subtract 3x – 2 5x – 2 2 2x + 5 The LCD is (2x + 5)(5x – 2), – 2x + 5 5x – 2 5x – 2 2x + 5 so multiply 3x – 2 by (5x – 2) 2x + 5 (5x – 2) 2 by (2x + 5) . and 5x – 2 (2x + 5) (3x – 2)(5x – 2) – 2(2x + 5) (2x + 5)(5x – 2) Subtract the numerators. 15x2 – 16x + 4 – (4x + 10) (2x + 5)(5x – 2) Multiply the binomials in the numerator. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 4a Continued 3x – 2 2 . Identify any x– 2x + 5 5x – 2 values for which the expression is undefined. Subtract 15x2 – 16x + 4 – 4x – 10 (2x + 5)(5x – 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 (2x + 5) and (5x – 2) equal 0. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 4b 2 + 64 2x x – 4 . Identify any xSubtract – 2 x – 64 x+8 values for which the expression is undefined. 2x2 + 64 x–4 – (x – 8)(x + 8) x+8 Factor the denominators. The LCD is (x – 3)(x + 8), 2x2 + 64 x–4 x–8 x–4 (x – 8) – (x – 8)(x + 8) x + 8 x – 8 so multiply x + 8 by (x – 8) . 2x2 + 64 – (x – 4)(x – 8) (x – 8)(x + 8) Subtract the numerators. 2x2 + 64 – (x2 – 12x + 32) (x – 8)(x + 8) Multiply the binomials in the numerator. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 4b 2 + 64 2x x – 4 . Identify any xSubtract – x2 – 64 x+8 values for which the expression is undefined. 2x2 + 64 – x2 + 12x – 32) Distribute the negative sign. (x – 8)(x + 8) x2 + 12x + 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 McDougal 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 McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Example 5A: 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 x2 + 7x + 10 (x – 1)(x – 3) x2 – 4x + 3 Holt McDougal Algebra 2 Multiply. 8-3 Adding and Subtracting Rational Expressions Example 5B: Simplifying Complex Fractions Simplify. Assume that all expressions are defined. 3 x + 2 x x–1 x Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator. 3 x (2x) + (2x) x 2 x – 1 (2x) x Holt McDougal Algebra 2 The LCD is 2x. 8-3 Adding and Subtracting Rational Expressions Example 5B Continued Simplify. Assume that all expressions are defined. (3)(2) + (x)(x) (x – 1)(2) Divide out common factors. x2 + 6 or 2(x – 1) Simplify. Holt McDougal Algebra 2 x2 + 6 2x – 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5a Simplify. Assume that all expressions are defined. x+1 x2 – 1 x x–1 Write the complex fraction as division. x+1 ÷ x x2 – 1 x–1 Write as division. x+1 x2 – 1 Multiply by the reciprocal. x–1 x Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5a Continued Simplify. Assume that all expressions are defined. x+1 x–1 (x – 1)(x + 1) x 1 x Holt McDougal Algebra 2 Factor the denominator. Divide out common factors. 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5b Simplify. Assume that all expressions are defined. 20 x–1 6 3x – 3 Write the complex fraction as division. 20 ÷ 6 x–1 3x – 3 Write as division. 20 3x – 3 x–1 6 Multiply by the reciprocal. Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5b Continued Simplify. Assume that all expressions are defined. 20 3(x – 1) x–1 6 10 Holt McDougal Algebra 2 Factor the numerator. Divide out common factors. 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5c Simplify. Assume that all expressions are defined. 1 1 + 2x 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 (2x)(x – 2) + (2x)(x – 2) x 2x x + 4 (2x)(x – 2) x–2 Holt McDougal Algebra 2 The LCD is (2x)(x – 2). 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 5c Continued Simplify. Assume that all expressions are defined. (2)(x – 2) + (x – 2) (x + 4)(2x) 3x – 6 or 3(x – 2) (x + 4)(2x) 2x(x + 4) Holt McDougal 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: 2d Let d represent the one-way distance. Total time: d + d 1.4 0.8 Use the formula t = d r . Average speed: Holt McDougal Algebra 2 2d d + d 1.4 0.8 The average speed is total distance . total time 8-3 Adding and Subtracting Rational Expressions Example 6 Continued 2d d = d = 5d and d = d = 5d . 1.4 7 7 0.8 4 4 5 5 2d(28) The LCD of the fractions in the 5d (28) + 5d (28) denominator is 28. 7 4 56d Simplify. 20d + 35d d + d 1.4 0.8 55d ≈ 1.0 55d Combine like terms and divide out common factors. The hiker’s average speed is 1.0 mi/h. Holt McDougal 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: 2d Let d represent the one-way distance. Total time: d + d 40 45 Use the formula t = d r . Average speed: Holt McDougal Algebra 2 2d d + d 40 45 The average speed is total distance . total time 8-3 Adding and Subtracting Rational Expressions Check It Out! Example 6 2d(360) d (360)+ d (360) 40 45 720d 9d + 8d 720d ≈ 42.4 17d 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 McDougal 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. 2x + 1 + x – 3 x–2 x+1 2. x x+4 – x2 + 36 x – 16 3x2 – 2x + 7 x ≠ –1, 2 (x – 2)(x + 1) x–9 x–4 x ≠ 4, –4 3. Find the least common multiple of x2 – 6x + 5 and x2 + x – 2. (x – 5)(x – 1)(x + 2) Holt McDougal Algebra 2 8-3 Adding and Subtracting Rational Expressions Lesson Quiz: Part II 4. Simplify defined. x+2 x2 – 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 McDougal Algebra 2