1 SOLVING RATIONAL EQUATIONS SIMPLIFYING RATIONAL EXPRESSIONS Standards 4, 7, 15, 25 ADDING RATIONAL EXPRESSIONS PROBLEM 1 RATIONAL EXPRESSIONS PROBLEM 2 FINDING THE LCD END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 4: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes. STANDARD 7 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator. STANDARD 15: Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is some-times true, always true, or never true. STANDARD 25: Students use properties from number systems to justify steps in combining and simplifying functions. ALGEBRA II STANDARDS THIS LESSON AIMS: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 ESTÁNDAR 4: Los estudiantes factorizan diferencias de cuadrados, trinomios cuadrados perfectos, y la suma y diferencia de dos cubos. ESTÁNDAR 7: Los estudiantes suman, restan, multiplican, dividen, reducen y evalúan expresiones racionales con monomios y polinomios en los denominadores y simplifican expresiones racionales, incluyendo aquellas con exponentes negativos en el denominador. ESTÁNDAR 15: Los estudiantes determinan si un estatuto algebraico específico involucrando expresiones racionales, expresiones radicales, o funciones logarítmicas o exponenciales es verdadero algunas veces, siempre o nunca. ESTÁNDAR 25: Los estudiantes usan propiedades de los sistemas numéricos para combinar y simplificar funciones. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 Simplify each expression: x + 3 2x x x x x + 15 x + 3 x -1 2 = 5(x+3) = x -1 2 x + 3 2( ) x -1 2 = 2 5 x x x - 6x = 27x x + 5 9x - 6x (3x+2)( ) 9x - 6x x + 5 9x - 6x (3x +2)(x + 5) (5) x (2) 3x x +3x +2 (5) + F O I L = 3x + 17x = 3x +15x + 2x = = = Standards 4, 7, 15, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 Find the Least Common Denominator for these denominators Standards 4, 7, 15, 25 8xy z 4 2 4x z 2 5 2x y xy z 4 2 4x z 2 5 = 2x y 32 =2 xy z =2 x z We choose each one of the different numbers and variables with the greatest exponent. 2 3 x 3 y 4 z 5 = 8 x 3 y 4 z 5 LCD: 7xy z x z 8 4 9x y z 57 We choose each one of the different numbers and variables with the greatest exponent. 3 3 x 8 y 7 z 6 = 189 x 8 y 7 z 6 LCD: 9x y z 57 7xy z x z 8 4 7xy z 5 6 = = 3 x y z = 3 x z PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 Add the following rational expressions: 3x x - 3 5x x + 3 x -9 2 x Find the LCD x - 3 x + 3 x -9 2 = (x+3)(x-3) = x x x - 3 5x x + 3 x -9 2 x x x - 3 5x x + 3 x -9 2 x = x + 3 x - 3 x x -7 3x +9x 2 5x - 15x = = 2 - (x – 7) 3x +9x 2 + x -9 2 = 2 - x + 7 5x - 15x 2 3x + 9x 2 + = x x + 3x - 15x + 9x - x + 7 = x x – 7x Get the same denominator for all fractions. Standards 4, 7, 15, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Solve the following rational equation: 6x x - 2 4x x + 2 x = Find the LCD x - 2 x + 2 x -4 2 = (x+2)(x-2) Multiply both sides by the LCD: 6x x - 2 4x x + 2 x = OR x -4 2 (x+2)(x-2) 6x x - 2 4x x + 2 x = (x+2)(x-2) x -4 2 (x-2)(4x) + (x+2)(6x) = 30 4x – 8x + 6x +12x = x + 4x = x + 4x – 30 = x + 2x – 15 = 0 Standards 4, 7, 15, 25 Note: This equation can’t have values at x=2, and x= -2, because with those values we have division by zero; which is not defined! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 We substitute values: x = -b b - 4ac 2a2a 2 +_ = -( ) ( ) - 4( )( ) 2( ) 2 +_ x = _ x = 10 +_ x = 10 +_ x = 10 x = x 1.54 x = x x = 10 x Using the Quadratic Formula: a= 5 b= 2 c= -15 = (-15) 10 +_ x 2 5x + 2x – 15 = 0 Standards 4, 7, 15, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 Solve the following rational equation: 7x x - 1 6x x + 1 x = Find the LCD x - 1 x + 1 x -1 2 = (x+1)(x-1) Multiply both sides by the LCD: 7x x - 1 6x x + 1 x = OR x -1 2 (x+1)(x-1) 7x x - 1 6x x + 1 x = (x+1)(x-1) x -1 2 (x-1)(6x) + (x+1)(7x) = 32 6x – 6x + 7x +7x = x + x = x + x – 32 = 0 2 Standards 4, 7, 15, 25 Note: This equation can’t have values at x=1, and x= -1, because with those values we have division by zero; which is not defined! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 We substitute values: x = -b b - 4ac 2a2a 2 +_ = -( ) ( ) - 4( )( ) 2( ) 2 +_ x = _ x = 26 +_ x = 26 +_ x = 26 x = x 1.5 x = x -1.6 x = 26 x Using the Quadratic Formula: a= 13 b= 1 c= -32 = (-32) 26 +_ x 2 13x + x – 32 = 0 Standards 4, 7, 15, 25 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved