I know how to solve the inequality x > (2/(x-1)) but how can I know not to mutiply all by (x-1)^2? That is a method for solving some inequalities.
you may multiply by provided you understand and you can deal algebraically with the resulting cubic expression.
In this case, I would prefer to use this method ...
critical values are , , and
these three values are where the expression on the left side of the inequality equals 0 or is undefined.
last step is to check a value from each interval between the critical values to see if the values in that interval make the original inequality true or false.
Suppose
Then if you multiply both sides of the inequality by
you would have to reverse the inequality.
This is because is negative when
but
The reversal happens if you change the sign of both sides,
which happens when you multiply both sides by a negative value,
or divide both sides by a negative value.
When you are dealing with inequalities, expressions involving x may be positive for certain x
and negative for other x.
Hence, if you multiply both sides by a square, you avoid that scenario (since real squares are non-negative).
Adding and subtracting the same value to both sides does not introduce any sign reversal,
nor does multiplying by 1, as shown by skeeter.