If x is not equal to 0 or -1, what is the value of |x-2|?

Official Explanation

In order to find the value of |x-2|, we first need to know the value of x. From question statement, we know that x is not equal to 0 or -1. Let's see if St. 1 and/or 2 can get us a unique value of x.

Analyzing St. 1 Independently

\(\frac{(|x+1|)}{(-|x|x^3 )}>0\)

Multiplying both sides by a positive number doesn’t impact the sign of inequality. So, multiplying both sides by \(\frac{(|x|)}{(|x+1|)}\), we get:

\(\frac{(-1)}{x^3} >0\)

(Note that we could multiply both sides by this number because we were told that x ≠ -1. Therefore, x + 1 ≠ 0 (division by 0 is not defined))

Now, multiplying both sides by -1 will reverse the sign of inequality. We get:

\(\frac{1}{x^3}<0\)

\(x^3\) will have the same +/- sign as x

This means, \(1/x\) is negative.

This means, x is negative.

This information is not sufficient to find the value of x.

Analyzing St. 2 independently
\(x^4 = 16\)

That is, \(x^4 - 2^4 = 0\)
\((x^2 + 2^2)(x+2)(x-2) = 0\)

This gives us, x = 2 or -2

Since we don’t get a unique value of x, not sufficient.

Combining St. 1 and St. 2

From St. 1, x is negative
From St. 2, x = 2 or -2

Combining both, x = -2

Sufficient.