Multiplying inequalities by x or ^2

The key idea is that when you multiply both sides of an inequality by a positive number, the inequality does not change direction. But if you multiply both sides by a negative number, then the inequality will change direction.

Now note that $x^2 \geq 0$ no matter what $x$ is. However, $x$ could be positive or negative, and this is why you have to be more careful when multiplying an inequality by $x$, or any other expression that could be both positive and negative for different values of $x$.

Let's solve the problem using both methods.

Method 1:
If you want to solve the inequality
\[ \frac{2}{x} < 1 \]
it is a good idea to multiply by $x^2$ so we don't need to change the direction to get
\[ 2 < x^2 \quad \Rightarrow \quad 0 < x^2 - 2x \quad \Rightarrow \quad 0 < x(x - 2), \]
and hence $x > 2$ or $x < 0$, or $x \in (-\infty, 0) \cup (2, \infty)$.

Method 2: If you multiply it by $x$, then we need to consider two cases:

Case I: $x > 0$, then the direction will not change and we get $2 < x$. Putting both conditions together we get $x > 2$.

Case II: $x < 0$, then the direction will change and we will get $2 > x$. Putting both conditions together we get $x < 0$.

Hence we get that $x > 2$ or $x < 0$, or $x \in (-\infty, 0) \cup (2, \infty)$.

You see that method 1 is much easier.