In inequality, real numbers are dealt with, imaginary or complex numbers are excluded because those numbers are not real: incomparable.
Thus, x≧0 and √x≧0 because if x<0, √x will be the imaginary or complex.
A. √x>-17: √x≧0, so √x>-17 Thus, A is solvable.
The set of solutions, in interval notation, is the interval [ 0, (+)infinity ).
B. √x>17: Since both sides of the inequality is positive, we can square them.
We have: x>289 Thus, B is solvable. The set of solutions is the interval ( 289, (+)infinity ).
C. √x<17: This can be restated: 0≦√x<17 Square both ends and the middle of inequality.
We have: 0≦x<289 Thus, C is solvable. The set of solutions is the interval [ 0, 289 )
D. √x<-17: This disagrees with the condition: √x≧0. Thus, D has no solutions.
The set of solutions is empty.
Therefore, the answer is D that has no solutions.