Is x > 0? (1) 1/x < 1 (2) 1/x > x

First of all, testing numbers is not the only way to solve this question.

The reason we wouldn’t multiply each side of 1/x < 1 by x is that we have no information on the sign of x. If x were positive, then we would obtain x > 1, as you noticed, but if x were negative, we would obtain x < 1 since we have to change the direction of an inequality whenever we multiply each side by a negative number.

To solve 1/x < 1, move the 1 on the right hand side to the left hand side:

1/x - 1 < 0

Now get the common denominator x and then combine the two fractions:

1/x - x/x < 0

(1 - x)/x < 0

Now, there are two ways this expression can be negative: Case 1: 1 - x is negative and x is positive, or Case 2: 1 - x is positive and x is negative.

For Case 1, we get 1 - x < 0 (which is equivalent to x > 1) and x > 0. The set of numbers that satisfy both of the inequalities x > 0 and x > 1 are x > 1. So, one way to satisfy the inequality 1/x < 1 is to pick values for x which are greater than 1.

For Case 2, we get 1 - x > 0 (which is equivalent to x < 1) and x < 0. The set of numbers that satisfy both inequalities are the set of x such that x < 0. Thus, the inequality 1/x < 1 can also be satisfied by picking a negative value for x.

As we can see, x could be either greater than 1 or less than 0. If we were to multiply each side of 1/x < 1 by x, we would completely miss the fact that negative values of x also satisfy the inequality 1/x < 1.

The explanation of why it’s wrong to rewrite 1/x > x as x^2 < 1 is similar to the explanation just presented.