To solve the inequality $x^2 - x < 0$, follow these steps:

Step 1: Factor the expression

We begin by factoring the quadratic expression: $x^2 - x = x(x - 1)$ So the inequality becomes: $x(x - 1) < 0$

Step 2: Determine critical points

The critical points of this inequality occur when $x(x - 1) = 0$, which gives: $x = 0 \quad \text{or} \quad x = 1$ These critical points divide the number line into three intervals: $(-\infty, 0)$, $(0, 1)$, and $(1, \infty)$.

Step 3: Analyze the sign of the product in each interval

We test the sign of $x(x - 1)$ in each interval:

• Interval $(-\infty, 0)$: Let $x = -1$. Substituting into $x(x - 1)$: $(-1)((-1) - 1) = (-1)(-2) = 2 > 0$ So, in this interval, $x(x - 1) > 0$.

• Interval $(0, 1)$: Let $x = \frac{1}{2}$. Substituting into $x(x - 1)$: $\left(\frac{1}{2}\right)\left(\frac{1}{2} - 1\right) = \left(\frac{1}{2}\right)\left(-\frac{1}{2}\right) = -\frac{1}{4} < 0$ So, in this interval, $x(x - 1) < 0$.

• Interval $(1, \infty)$: Let $x = 2$. Substituting into $x(x - 1)$: $(2)(2 - 1) = (2)(1) = 2 > 0$ So, in this interval, $x(x - 1) > 0$.

Step 4: Solution

We are looking for where $x(x - 1) < 0$, which occurs in the interval $(0, 1)$.

Thus, the solution to the inequality is: $\boxed{(0, 1)}$

Would you like further details or explanations on any of the steps?

Here are some related questions:

1. How do you solve quadratic inequalities graphically?
2. What is the difference between solving $x^2 - x \leq 0$ and $x^2 - x < 0$?
3. How do critical points help in solving inequalities?
4. Can you solve similar inequalities with different factorable quadratics?
5. How would the solution change if the inequality were $x^2 - x > 0$?

Tip: Always consider the sign changes at critical points when solving polynomial inequalities.