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Rational Equations Solving Rational Equations

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8/2/2013Rational Equations2 Definition f(x) is a rational function if and only if f(x) = Example 1. f(x) = Rational Functions p(x) q(x) where p(x) and q(x) are polynomial 3x 2 + 4x + 1 x 3 – 1 functions with q(x) 0

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8/2/2013Rational Equations3 Solving Rational Equations How do we solve equations of form: Method 1: Clear Fractions = x + 2 = 23 x (x + 2) (x + 2) 23 = 15x + 30 –7 = 15x = –7 15 x Solution Set: { } 7 15 – = g(x) f(x) h(x) 1. Solve:

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8/2/2013Rational Equations4 Method 1: Clear Fractions 2. Solve: Solving Rational Equations = x + 1 x + 2 x + 5 x + 7 (x + 2)(x + 7) = x + 5 x + 7 x + 1 x + 2 (x + 2)(x + 7) (x + 5)(x + 2) = (x + 1)(x + 7) x 2 + 7x + 10 = x 2 + 8x + 7 7x + 10 = 8x = x Solution Set: { 3 }

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8/2/2013Rational Equations5 Method 2: Cross Multiplication Basic Principle: 1. Solve: Solving Rational Equations = a b c d if and only if ad = bc = x + 1 x + 2 x + 5 x + 7 (x + 2)(x + 5) = (x + 1)(x + 7) x 2 + 7x + 10 = x 2 + 8x + 7 7x + 10 = 8x = x Solution Set: { 3 }

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8/2/2013Rational Equations6 Method 2: Cross Multiplication 2. Solve: Cross multiplying Solving Rational Equations = x – x = (x – 3)(x + 3) 7 = x 2 – 9 0 = x 2 – = x2x2 Zero Product Property Square Root Property

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8/2/2013Rational Equations7 Method 2: Cross Multiplication 2. Solve: Solving Rational Equations 0 = (x + 4)(x – 4) 0 = x = x – 4 OR –4 = x 4 = x Solution Set: { – 4, 4 } = x ± 4 = x – x = x 2 – 9 0 = x 2 – = x2x2 Zero Product Property Square Root Property = x2x2 √ ± 16 √

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8/2/2013Rational Equations8 Solving Rational Equations Method 2: Cross Multiplication 3. Solve: = x + 1 5x – (5x – 3) = 3(x + 1) 10x – 6 = 3x + 3 7x = 9 = 9 7 x Solution Set: 9 7 { }

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8/2/2013Rational Equations9 Method 3: Graphical Approach 1. Solve: Solving Rational Equations x + 1 x – 5 = 2 and y 2 = 2 x y –2 –3 y1y1 Vertical Asymptote x = 5 y2y2 Horizontal Asymptote y = 1 Intersection at (11, 2) (11, 2) Hence: x = 11 y 1 = y 2 ? For what x is this true ? y 1 intercepts : Horizontal : ( –1, 0 ) Vertical : ( 0, –1/5 ) Let y 1 x + 1 x – 5 =

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8/2/2013Rational Equations10 Think about it !

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