Dear Mathematicians, Control Engineers, and Optimization Enthusiastics, Could you please compute the domain through which this inequality holds (3/(x-2)) -1, and please don’t use any symbolic math software)
(-1, 2)
Proof: consider the three cases (x - 2) > 0, (x - 2) < 0, and (x - 2) = 0.
If (x - 2) > 0, reformulating the inequality leads to -1 > x, which together with the initial assumption (x - 2) > 0 combines to the empty set.
If (x - 2) < 0, reformulating the inequality leads to -1 < x, which together with the initial assumption (x - 2) < 0 combines to -1 < x < 2.
If (x - 2) = 0, we can not even say that 3 / (x - 2) is "minus infinity", since its sign changes at this pole.
The answer is -1 < x < 2 as follows
if x-2 > 0 then
3 < -1 (x-2) making x < -1 which is not possible
if x-2 < 0 then
3> -1 (x-2) making x > -1 hence x < 2 and x > -1.
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