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Ejercicio 138

Gestor Baldor
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CAPITULO XIV

Operaciones con Fracciones
Ejercicio 138
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  1. 1+ x+1 x–1 1 x–1 – 1 x+1 = x– 1 +x+ 1 x–1 x +1– x +1 ( x+1 ) ( x–1 ) = 2 x( x+1 ) 2 =x( x+1 )
  2. 1 x–1 + 2 x+1 x–2 x + 2x+6 x+1 = x+1+2( x–1 ) ( x+1 ) ( x–1 ) ( x–2 ) ( x+1 ) +x( 2x+6 ) x ( x+1 ) = x+1+2x–2 x–1 x 2 –x–2+2 x 2 +6x x = 3x–1 x–1 3 x 2 +5x–2 x = 3x–1 x–1 3 x 2 +6x–x–2 x = 3x–1 x–1 3x( x+2 ) –( x+2 ) x = 3x–1 x–1 ( 3x–1 ) ( x+2 ) x = x ( x–1 ) ( x+2 )
  3. a a–b – b a+b a+b a–b + a b = a( a+b ) –b( a–b ) ( a–b ) ( a+b ) b( a+b ) +a( a–b ) b ( a–b ) = a 2 + ab – ab + b 2 a+b ab + b 2 + a 2 – ab b = b ( a 2 + b 2 ) ( a 2 + b 2 ) ( a+b ) = b a+b
  4. x+3 x+4 – x+1 x+2 x–1 x+2 – x–3 x+4 = ( x+3 ) ( x+2 ) –( x+4 ) ( x+1 ) ( x+4 ) ( x+2 ) ( x–1 ) ( x+4 ) –( x–3 ) ( x+2 ) ( x+4 ) ( x+2 ) = x 2 +5x+6–( x 2 +5x+4 ) x 2 +3x–4–( x 2 –x–6 ) = x 2 + 5x +6– x 2 – 5x –4 x 2 +3x–4– x 2 +x+6 = 2 4x+2 = 2 2 ( 2x+1 ) = 1 2x+1
  5. m 2 n – m 2 – n 2 m+n m–n n + n m = m 2 ( m+n ) –n( m 2 – n 2 ) n ( m+n ) m( m–n ) + n 2 n m = m 3 + m 2 n – m 2 n + n 3 m+n m 2 –mn+ n 2 m = m ( m 3 + n 3 ) ( m+n ) ( m 2 –mn+ n 2 ) =m
  6. a 2 b 3 + 1 a a b – b–a a–b = a 3 + b 3 a b a( a–b ) –b( b–a ) b ( a–b ) = a 3 + b 3 a b 2 a 2 – ab – b 2 + ab a–b = ( a 3 + b 3 ) ( a–b ) a b 2 ( a 2 – b 2 ) = ( a+b ) ( a 2 –ab+ b 2 ) ( a–b ) a b 2 ( a 2 – b 2 ) = a 2 –ab+ b 2 a b 2
  7. 1+ 2x 1+ x 2 2x+ 2 x 5 +2 1– x 4 = 1+ x 2 +2x 1+ x 2 2x( 1– x 4 ) +2 x 5 +2 1– x 4 = ( x+1 ) 2 1+ x 2 2x– 2 x 5 + 2 x 5 +2 ( 1+ x 2 ) ( 1– x 2 ) = ( x+1 ) 2 ( 1– x 2 ) 2x+2 = ( x+1 ) 2 ( 1– x 2 ) 2 ( x+1 ) = ( x+1 ) ( 1– x 2 ) 2
  8. x+y x–y – x–y x+y x+y x – x+2y x+y = ( x+y ) 2 – ( x–y ) 2 ( x–y ) ( x+y ) ( x+y ) 2 –x( x+2y ) x ( x+y ) = [ ( x+y ) –( x–y ) ] [ ( x+y ) +( x–y ) ] x–y x 2 + 2xy + y 2 – x 2 – 2xy x = ( x +y– x +y ) ( x+ y +x– y ) x–y y 2 x = ( 2 y ) ( 2x ) x–y y 2 x = 4 x 2 y( x–y )
  9. a+x a–x – b+x b–x 2 a–x – 2 b–x = ( a+x ) ( b–x ) –( a–x ) ( b+x ) ( a–x ) ( b–x ) 2( b–x ) –2( a–x ) ( a–x ) ( b–x ) = ab–ax+bx– x 2 –( ab+ax–bx– x 2 ) 2( b– x –a+ x ) = ab –ax+bx– x 2 – ab –ax+bx+ x 2 2( b–a ) = 2bx–2ax 2( b–a ) = 2 x ( b–a ) 2 ( b–a ) =x
  10. a a+x – a 2a+2x a a–x + a a+x = a a+x [ 1– 1 2 ] a[ 1 a–x + 1 a+x ] = a a+x [ 2–1 2 ] a[ a+ x +a– x ( a+x ) ( a–x ) ] = a 2 ( a+x ) 2 a 2 ( a+x ) ( a–x ) = a–x 4a
  11. a+2b a–b + b a a+b a + 3b a–b = a( a+2b ) +b( a–b ) a( a–b ) ( a+b ) ( a–b ) +3ab a( a–b ) = a 2 +2ab+ab– b 2 a 2 – b 2 +3ab = a 2 +3ab– b 2 a 2 +3ab– b 2 =1
  12. 1– 7 x + 12 x 2 x– 16 x = x 2 –7x+12 x 2 x 2 –16 x = ( x–3 ) ( x–4 ) x ( x–4 ) ( x+4 ) = x–3 x( x+4 )
  13. a 2 b – b 2 a 1 b + 1 a + b a 2 = a 3 – b 3 a b a 2 +ab+ b 2 a 2 b = a( a 3 – b 3 ) a 2 +ab+ b 2 = a( a–b ) ( a 2 +ab+ b 2 ) a 2 +ab+ b 2 =a( a–b )
  14. x–2y– 4 y 2 x+y x–3y– 5 y 2 x+y = ( x–2y ) ( x+y ) –4 y 2 x+y ( x–3y ) ( x+y ) –5 y 2 x+y = x 2 –xy–2 y 2 –4 y 2 x 2 –2xy–3 y 2 –5 y 2 = x 2 –xy–6 y 2 x 2 –2xy–8 y 2 = x 2 +2xy–3xy–6 y 2 x 2 –4xy+2xy–8 y 2 = x( x+2y ) –3y( x+2y ) x( x–4y ) +2y( x–4y ) = ( x–3y ) ( x+2y ) ( x+2y ) ( x–4y ) = x–3y x–4y
  15. 2 1–a + 2 1+a 2 1+a – 2 1–a = 2 [ 1 1–a + 1 1+a ] 2 [ 1 1+a – 1 1–a ] = 1+ a +1– a ( 1–a ) ( 1+a ) 1–a–( 1+a ) ( 1–a ) ( 1+a ) = 2 1 –a– 1 –a =– 2 2 a =– 1 a
  16. 1 x+y+z – 1 x–y+z 1 x–y+z – 1 x+y+z = x–y+z–( x+y+z ) ( x+y+z ) ( x–y+z ) x+y+z–( x–y+z ) ( x+y+z ) ( x–y+z ) = x –y+ z – x –y– z x +y+ z – x +y– z =– 2y 2y =–1
  17. 1+ 2b+c a–b–c 1– c–2b a–b+c = a–b– c +2b+ c a–b–c a–b+c–( c–2b ) a–b+c = a+b a–b–c a–b+ c – c +2b a–b+c = a+b a–b–c a+b a–b+c = a–b+c a–b–c
  18. a 1–a + 1–a a 1–a a – a 1–a = a 2 + ( 1–a ) 2 a( 1–a ) ( 1–a ) 2 – a 2 a( 1–a ) = a 2 +1–2a+ a 2 1–2a+ a 2 – a 2 = 2 a 2 –2a+1 1–2a
  19. x+1– 6x+12 x+2 x–5 x–4+ 11x–22 x–2 x+7 = ( x+1 ) ( x+2 ) –( 6x+12 ) x+2 x–5 ( x–4 ) ( x–2 ) +11x–22 x–2 x+7 = x 2 –3x–10 x+2 x–5 x 2 +5x–14 x–2 x+7 = ( x–5 ) ( x+2 ) x+2 x–5 ( x+7 ) ( x–2 ) x–2 x+7 =1
  20. 1 1+ 1 x = 1 x+1 x = x x+1
  21. 1 1+ 1 1– 1 x = 1 1+ 1 x–1 x = 1 1+ x x–1 = 1 x–1+x x–1 = 1 2x–1 x–1 = x–1 2x–1
  22. 1– 1 2+ 1 x 3 –1 =1– 1 2+ 1 x–3 3 =1– 1 2+ 3 x–3 =1– 1 2x–6+3 x–3 =1– x–3 2x–3 = 2x–3–( x–3 ) 2x–3 = 2x– 3 –x+ 3 2x–3 = x 2x–3
  23. 2 1+ 2 1+ 2 x = 2 1+ 2 x+2 x = 2 1+ 2x x+2 = 2 x+2+2x x+2 = 2 3x+2 x+2 = 2( x+2 ) 3x+2
  24. 1 x– x x– x 2 x+1 = 1 x– x x( x+1 ) – x 2 x+1 = 1 x– x x 2 +x– x 2 x+1 = 1 x– x ( x+1 ) x = 1 x – x –1 =–1
  25. 1 a+2– a+1 a– 1 a = 1 a+2– a+1 a 2 –1 a = 1 a+2– a( a+1 ) a 2 –1 = 1 a+2– a ( a+1 ) ( a+1 ) ( a–1 ) = 1 a+2– a a–1 = 1 ( a+2 ) ( a–1 ) –a a–1 = 1 a 2 + a –2– a a–1 = a–1 a 2 –2
  26. x–1 x+2– x 2 +2 x– x–2 x+1 = x–1 x+2– x 2 +2 x( x+1 ) –( x–2 ) x+1 = x–1 x+2– x 2 +2 x 2 + x – x +2 x+1 = x–1 x+2– x 2 +2 x 2 +2 x+1 = x–1 x+2–( x+1 ) = x–1 x +2– x –1 =x–1
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