Let $x$ be a real number.  Consider the following five statements:

$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$

What is the maximum number of these statements that can be true for any value of $x$?
Let's graph $y = x^2$ and $y = x - x^2$ on the same coordinate axis. And let's also graph the lines $y = 1$ and $y = 0$, since the statements involve inequalities. The second statement says that $x$ is outside the interval $(-1, 1)$. So that means that either $x < -1$ or $x > 1$. Right. Which means that the third and fourth statements cannot both be true. That's true. The third statement says $-1 < x < 0$, and the fourth statement says $0 < x < 1$. So, the maximum number of statements that can be true at the same time is $3$.

