Question

Which Is a Solution to the Inequality?

Answer 1

dont be tempted to cross-multiply with these
x - 6 > x -4 will give you an incorrect solution

Answer 2

hehe, right, I gather the issue is, "what values can 'x' take and no render undefined"?

Answer 3

@zairhenrique

Answer 4

one way is to simply try out values

hint try plugging in x = 5

Answer 5

\[\frac{ 1 }{ x-4 } > \frac{ 1 }{ x-6 } \iff \frac{ 1 }{ x-4 } - \frac{ 1 }{ x-6 } >0\]\[\frac{ (x-6) - (x-4) }{ (x-4)(x-6) } > 0 \iff \frac{ -6+4 }{ (x-4)(x-6) }>0\]\[\frac{ -2 }{ (x-4)(x-6) } > 0\]So,\[(x-4)(x-6) <0\]Possibilities:
(i)\[x-4<0\]and\[x-6>0\]\[x<4\]and\[x>6\](impossible).
(ii)
\[x-4>0\]and\[x-6<0\]\[x>4\]\[x<6\]Possible!
Solution:\[4<x<6\]