问一道数学题 当x>2时,试比较分式(x-2)/(x-1)和(x-3)/(x-2)的值的大小

问题描述:

问一道数学题 当x>2时,试比较分式(x-2)/(x-1)和(x-3)/(x-2)的值的大小

(x-2)/(x-1)
=(x-2)(x-2)/(x-1)(x-2)
=(x^2-4x+4)/(x-1)(x-2)
(x-3)/(x-2)
=(x-3)(x-1)/(x-1)(x-2)
=(x^2-4x+3)/(x-1)(x-2)
因为x>2,
所以x^2-4x+4>x^2-4x+3
(x-1)(x-2)>0
所以(x-2)/(x-1)>(x-3)/(x-2)

(x-2)/(x-1)>(x-3)/(x-2)

(x-2)/(x-1)
=(x-1-1)/(x-1)
=(x-1)/(x-1)-1/(x-1)
=1-1/(x-1)
(x-3)/(x-2)
=(x-2-1)/(x-2)
=(x-2)/(x-2)-1/(x-2)
=1-1/(x-2)
x>2
所以x-1>x-2>0
所以1/(x-1)-1/(x-2)
-1/(x-1)>1-1/(x-2)
所以(x-2)/(x-1)>(x-3)/(x-2)