Question

Please help me solve: 2/(2x-5)<1/(x+7)

Answer 1

be very very careful here

Answer 2

you do not know the sign of \(x\) so you cannot do the usual stuff like "cross multiply" or flip them
you have to start with
\[\frac{2}{2x-5}-\frac{1}{x+7}<0\] then solve

Answer 3

satellite showed you how to start. It is *always* safest to add or subtract when solving relations.

once you get
\[ \frac{2}{2x-5}-\frac{1}{x+7}<0 \]
add the two fractions to get
\[ \frac{ 2x+14-2x + 5}{(2x-5)(x+7)} < 0 \]
the top simplifies to 19:
\[ \frac{ 19}{(2x-5)(x+7)} < 0 \]

now ask yourself: when is the left side negative ? (i.e. less than zero)

it is negative if (2x-5) is positive and *at the same time* x+7 is negative (because + times - is -). so you need
2x-5 > 0 and x+7<0

or (2x-5) is negative and *at the same time* x+7 is positive
2x-5 < 0 and x+7> 0

can you finish ?

Answer 4

Thank you ever so much. That is just what I needed!