Solving Rational Equations Solving Rational Equations 8/2/2013 Rational Equations

Solving Rational Equations Rational Functions Rational Functions and Models Definition f(x) is a rational function if and only if f(x) = Example 1. f(x) = p(x) q(x) where p(x) and q(x) are polynomial functions with q(x)  0 Rational Functions: Definition The most common error students make in identifying rational functions is assuming that any quotient function is a rational function. Care must be taken to identify each of the numerator and denominator as a polynomial. No other type of quotient function is a rational function. Note that some rational functions can be reduced to simple linear functions, provided the zeros of the denominator are excluded. This is the case in Example 2. Note that not all quotient type functions are rational functions, as is illustrated in Example 3. The fractional power in the numerator indicates the cube of a square root of x. Since polynomials must have integer powers in all terms, the numerator is not a polynomial and hence the quotient f(x) is not a rational function. 3x2 + 4x + 1 x3 – 1 8/2/2013 Rational Equations Section 4.5 v5.0 2 10/11/2012 8/2/2013 Rational Equations

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

Solving Rational Equations Method 1: Clear Fractions 2. Solve: = x + 1 x + 2 x + 5 x + 7 x + 1 x + 2 (x + 2)(x + 7) (x + 2)(x + 7) = x + 5 x + 7 (x + 5)(x + 2) = (x + 1)(x + 7) x2 + 7x + 10 = x2 + 8x + 7 7x + 10 = 8x + 7 3 = x Solution Set: { 3 } 8/2/2013 Rational Equations

Solving Rational Equations Method 2: Cross Multiplication Basic Principle: 1. Solve: = 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) x2 + 7x + 10 = x2 + 8x + 7 7x + 10 = 8x + 7 3 = x Solution Set: { 3 } 8/2/2013 Rational Equations

Solving Rational Equations Method 2: Cross Multiplication 2. Solve: Cross multiplying = x – 3 7 1 x + 3 7 = (x – 3)(x + 3) 7 = x2 – 9 Zero Product Property Square Root Property = x2 – 16 16 = x2 8/2/2013 Rational Equations

Solving Rational Equations Method 2: Cross Multiplication 2. Solve: = x – 3 7 1 x + 3 7 = x2 – 9 Zero Product Property Square Root Property = x2 – 16 16 = x2 = (x + 4)(x – 4) = x2 √ ± 16 = x + 4 OR = x – 4 = x ± 4 –4 = x 4 = x Solution Set: { – 4, 4 } 8/2/2013 Rational Equations

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

Solving Rational Equations Method 3: Graphical Approach 1. Solve: x + 1 x – 5 = 2 Hence: x = 11 x y 3 6 9 11 –2 –3 Let y1 x + 1 x – 5 = y1 Horizontal Asymptote y = 1 and y2 = 2 y2 (11, 2) y1 = y2 ?  For what x is this true ?   Vertical Asymptote x = 5 y1 intercepts : Intersection at (11, 2) Horizontal : ( –1, 0 ) Vertical : ( 0, –1/5 ) 8/2/2013 Rational Equations 8/2/2013 Rational Equations

Think about it ! 8/2/2013 Rational Equations