If x ≠ 0, is |x| < 1 ? (1) x^2 < 1 (2) |x| < 1/x

Yes, here is an excerpt from my Inequalities book:

How to Handle an Inequality with Multiple Factors

Consider any inequality of the form (x - a)(x - b)(x - c) < 0. For clarity, let’s work with the example x(x - 3) < 0 discussed above.

Step 1: Make a number line and plot the points a, b and c on it. In our example, a = 0 and b = 3. The number line is divided into sections by these points. In our example, it is divided into 3 sections – ‘greater than 3’, ‘between 0 and 3’ and ‘less than 0.’

Step 2: Starting from the rightmost section, mark the sections with alternate positive and negative signs. The rightmost section will be positive, the next one will be negative, the next one will be positive and so on... The inequality will be positive in the sections where we have the positive signs and it will be negative in the sections where we have the negative signs.



Therefore, x*(x - 3) will be positive in the ranges ‘x > 3’ and in ‘x < 0’.
It will be negative in the range ‘0 < x < 3’.
Hence, the values of x for which x*(x - 3) < 0 is satisfied is 0 < x < 3.


Explanation:

When we plot the points on the line, the number line is divided into various sections. Values of x in the right most section will always give us positive value of the expression. The reason for this is that if x > 3, all factors will be positive i.e. x and (x - 3), both will be positive.

When we jump to the next region i.e. between x = 0 and x = 3, the values of x will give us negative values for the entire expression because now, only one factor, (x - 3), will be negative. All other factors will be positive.

When we jump to the next region on the left where x < 0, expression will be positive again because now both factors x and (x - 3) are negative. The product of two negatives is positive so the expression will be positive.
Similarly, we can solve a question with any number of factors.