# Remember a a c a d b c b d b c d ﾹ ﾹ 1 2 3x 2 x

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Remember : a a c a d b c b d b c d = = 1. 2 3x 2 x 6x 4 x 7x 4 2 4 4 4 5x 7 5x 7 4 6 6 + + + + + = = 3 6 21x 12 5x 7 10x 14 + = , x 7 5 2. 2 3 1 9 2x 9 2x 3x 9 2x 2x 3 6x 6x 2x 7 2x 7 6x 2x 7 6x 3x 3x + + + + = = = 1 3x 9 2x 2x 7 4x 14 + = 3. Watch negatives in numerator and denominator. 2 2 2 2 2 2 2 2 5x 3 2x 5x 3 2x 5x 3 x 2x 5x 3 3 2 2 2 2 8x 3 3x 8x 3 3x 8x 3 2 3x 8x 3 x 3 3 3 3 + + = = = + + ( ) ( ) 2x 1 x 3 + = ( ) ( ) 3 2 3x 1 x 3 + ( ) ( ) 3 2x 1 6x 3 or 2 3x 1 6x 2 + + = + + ASSIGNMENT: Page 336 10a–d All complex equations. M20–1 LP Ch 6 Ratn Exp (Chandler).doc 12
Day 6 Rational Equation – An equation that contains one or more rational expressions ie. 2 3 3 8 6 2 , 0 , 4 7 9 10 x x x x x x = = = + Are all rational expressions ***NOTE*** Working with rational equations is much the same as working with rational expressions, several differences are: In an equation whatever operation we perform on one side (of the equal sign) we must also perform on the other side In an equation we can clear fractions (rational expressions) by multiplying each term by the lowest common denominator (L.C.D) To solve a rational equation we simply, Factor all expressions Identify and state any non–permissible values (N.P.V.’s) Multiply each expression by the lowest common denominator (L.C.D.)to clear fractions Simplify remaining expression Solve expression by isolating variable to one side of our equation Check our answers (roots) for extraneous roots Recall An Extraneous Root is a number (value) obtained when solving an equation that does not however satisfy the initial restrictions on the variable or satisfy the original equation I. Equations with rational expressions. 1. x 5 5x 3 2 6 + = 2. 1 8 1 x x 1 x 5 15 3 + + = + 2x 5 5x 6 6 3 2 6 � � + = � � � � ( ) 1 8 1 15 x x 1 x 15 5 15 3 + + = + 2x + 15 = 5x 3x + 8 + 5x = 15 + 15x 15 = 3x 8x + 8 = 15 + 15x 5 = x 7 = 7x 1 = x 3. x 3 x 3x 7 4 6 2 + = 4. ( ) 2 2 x 5 2x 3 9 + + = x 3 x 3x 7 12 12 4 6 2 + � � = � � � � ( ) ( ) 2 x 5 2 9 2x 9 3 9 + + = 3x + 9 2x = 18x – 42 6x + 30 + 2 = 18x 51 = 17x 32 = 12x 3 = x 8 3 = x M20–1 LP Ch 6 Ratn Exp (Chandler).doc 13
5. ( ) 1 x 1 x 6 4 6 5 + + = 6. 3x 7 5x 4 2 3 = one rational expression x 6 x x 8 60 60 4 6 5 + + � � + = � � � � 9x – 21 = 10x 8 on each side, cross 15x + 90 + 10x = 12x + 96 13 = x multiply 13x = 6 x = 6 13 7. 4 6 2 x + = , x 0 8. 5 3 x 12 2x x 5x + + = , x 0 ( ) 4 x 6 2 x x + = 5 3 x 12 10x 10x 2x x 5x + � � + = � � � � 4 + 6x = 2x 25 + 30 = 2x + 24 4x = 4 31 = 2x x = 1 15.5 = x 9. 3 5 2 x 2 x 7 x 2 + = + ( ) ( ) ( ) ( ) 3 5 2 x 2 x 7 x 2 x 7 x 2 x 7 x 2 � � + + = + � � + � � 3x + 21 + 5x – 10 = 2x + 14 6x = 3 x = 1 2 M20–1 LP Ch 6 Ratn Exp (Chandler).doc 14
( ) ( ) 2 2 2 2 1 2 15 3 5 2( 1) 2 1 ( 5)( 3) 3 5 2( 1) ( 5) x x x x x x x x x x x x + + = + + + + = + + + + ( 3) ( 5) x x + ( ) ( ) 2( 5)( 3) 1( 5)( 3) 3 5 ( 3) 2( 1) 2( 5) 1( 3) 2 2 2 10 3 4 12 3 3 15 5 , 5 ( . . .) x x x x x x x x x x x x x x x x x however x N PV No Solution + + + = + + + + = + + + = + = = = \ 10.