Inequality Challenge 144

For option 1.: Let's assume $A + 2 = B + 2$

which implies that $A = B$

But that is a $Contradiction$

So our assumption is incorrect, so $A$ isn't equal to $B$ . Option 1. is $Correct$ .

For option 3.: Let's assume $A^{3} = B^{3}$

which implies $A = B$ ( By taking cube roots on both sides). Here note that cube root of positive real number is positive and cube root of negative real number is negative.

Which is again a Contradiction! So option 3. is $Correct$ .

Counter example for option 2.: Let $A = 2$ , $B = -2$ .

Clearly both aren't equal.

But $A^{2} = 2^{2} = 4$ and $B^{2} = (-2)^{2} = 4$ , so $A^{2} = B^{2}$ , so option 2. is $Wrong$ .