Algebra - inequality problem

GRE Quant Problem Solving
1

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For this question, how do you get the solution of -1? If you solve it, it’s X^2 = 1

Is this a scenario where to solve this question, you plug in numbers to test which inequality hold? For example

X < 1 is false because if x = 0, the inequality wouldn’t hold
-1 < x < 1 is false because if x = 0, the inequality wouldn’t hold again given the range contains 0
-1 < x < 0 is false because if x = -0.5, x^3 > x
x < -1 is true because of x = -2, x^3 will be more negative (further from x = -2, eg -8)
0 < x < 1 is true (special property for a number between 0 and 1) - x^2 < x < square root x

2

Not quite. I presume you’re dividing by x on both sides, but notice that x can be negative, in which case the inequality would flip.

This is in general a very good way to handle these kinds of problems.