Question

a)$\frac{ x^{2}-7x+12 }{ x^{2} -16}$ · $\frac{x-2}{x-3}$ + $\frac{2x+14}{x+4}$=?
b)$\frac{2}{x-2} + \frac{x+2}{x+1}$ · $( \frac{x-1}{x+2} - \frac{6}{x-2})$=?
c)$\frac{x+2}{x+1}$ · $( \frac{x+1}{x+2} - \frac{ x^{2}-1 }{ x^{2} -4} ) + \frac{x-1}{x-2} =?$
d)$\frac{5x+6}{ x^{2} -4} - \frac{ x+3}{x-2} : \frac{ x^{2}+5x+6 }{x-2} = ?$
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Answer 1

a)
(x²-7x+12)/(x²-16) ×(x-2)/(x-3) +(2x+14)/(x+4) =
=(x²-4x-3x+12)/(x+4)(x-4) ×(x-2)/(x-3) +(2x+14)/(x+4) =
=[x(x-4)-3(x-4))/(x+4)(x-4) ×(x-2)/(x-3) +(2x+14)/(x+4) =
=(x-4)(x-3)/(x+4)(x-4) ×(x-2)/(x-3) +(2x+14)/(x+4) =
=(x-2)/(x+4) +(2x+14)/(x+4) =
=(3x+12)/(x+4)=
=3(x+4)/(x+4)=3


b)
2/(x-2)+(x+2)/(x+1)×[(x-1)/(x+2) - 6/(x-2)]=
=2/(x-2)+(x+2)/(x+1)×[(x-1)(x-2) - 6(x+2)]/(x²-4)=
=2/(x-2)+(x+2)/(x+1)×(x²-1x-2x+2 - 6x-12)/(x²-4)=
=2/(x-2)+(x+2)/(x+1)×(x²-9x-10)/(x²-4)=
=2/(x-2)+(x+2)/(x+1)×(x²-10x+x-10)/(x²-4)=
=2/(x-2)+(x+2)/(x+1)×[x(x-10)+(x-10)]/(x²-4)=
=2/(x-2)+(x+2)/(x+1)×(x-10)(x+1)/(x+2)(x-2)=
=2/(x-2)+(x-10)/(x-2)=
=(2+x-10)/(x-2)=(x-8)/(x-2)

c) (x+2)/(x+1) ×[(x+1)/(x+2) -(x²-1)/(x²-4)]+(x-1)/(x-2)=
=(x+2)/(x+1) ×[(x+1)(x-2) -(x²-1)]/(x²-4)+(x-1)/(x-2)=
=(x+2)/(x+1) ×(x²+x-2x-2 -x²+1)/(x+2)(x-2)+(x-1)/(x-2)=
=(x+2)/(x+1) ×(-x-1)/(x+2)(x-2)+(x-1)/(x-2)=
=(x+2)/(x+1) ×[-(x+1)]/(x+2)(x-2)+(x-1)/(x-2)=
=-1/(x-2)+(x-1)/(x-2)=
=(-1+x-1)/(x-2)=
=(x-2/(x-2)=1

d)
(5x+6)/(x²-4) -(x+3)/(x-2) : (x²+5x+6)/(x-2)=
=(5x+6)/(x²-4) -(x+3)/(x-2) ×(x-2)/(x²+2x+3x+6)=
=(5x+6)/(x²-4) -(x+3)/(x²+2x+3x+6)=
=(5x+6)/(x+2)(x-2) -(x+3)/ [x(x+2)+3(x+2)]=
=(5x+6)/(x+2)(x-2)- (x+3)/(x+3)(x+2)=
=(5x+6)/(x+2)(x-2)- 1/(x+2)=
=(5x+6- x+2)/(x+2)(x-2)=
=(4x+8)/(x+2)(x-2)=
=4(x+2)/(x+2)(x-2)=
=4/(x-2)
=(5x+6-(x²+3x+2x+6)]/(x+2)(x-2) × (x-2)/(x+2)(x+3)=
=(5x+6-x²-3x-2x-6)]/(x+2)(x-2) × (x-2)/(x+2)(x+3)=
=(-x²)/(x+2) × 1/(x+2)(x+3)=