We're given that -1<x<0 or 0<x<1. To see if each statement must be true, we should test these ranges.
I.\(x^2<|x|\)
If x is negative, then |x|=-x, so expressionbecomes
\(x^2<|x|\)
\(x^2<-x\)
\(x>-1\) (dividing by x and reversing the inequality since x is negative)
So if x is negative, -1<x<0, which falls inside the negative part of the given range for x.
If x is positive, then |x|=x, so expression becomes
\(x^2<|x|\)\(x^2<x\)
\(x<1\) So if x is positive, 0<x<1, which falls
...
I.\(x^2<|x|\)
If x is negative, then |x|=-x, so expressionbecomes
\(x^2<|x|\)
\(x^2<-x\)
\(x>-1\) (dividing by x and reversing the inequality since x is negative)
So if x is negative, -1<x<0, which falls inside the negative part of the given range for x.
If x is positive, then |x|=x, so expression becomes
\(x^2<|x|\)\(x^2<x\)
\(x<1\) So if x is positive, 0<x<1, which falls
...
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