We need to solve the inequality:

$x(x - 1)(x - 2) > 0$

Step 1: Find the critical points

The expression is zero when:

$x(x - 1)(x - 2) = 0$

Solving for $x$, we get the critical points:

$x = 0, x = 1, x = 2$

These points divide the number line into four intervals:

1. $(-\infty, 0)$
2. $(0, 1)$
3. $(1, 2)$
4. $(2, \infty)$

Step 2: Test each interval

We pick a test point in each interval and check the sign of $x(x - 1)(x - 2)$.

• Interval $(-\infty, 0)$, choose $x = -1$:
  $(-1)(-1 - 1)(-1 - 2) = (-1)(-2)(-3) = -6 \quad (\text{negative})$

• Interval $(0, 1)$, choose $x = \frac{1}{2}$:
  $\left( \frac{1}{2} \right) \left( \frac{1}{2} - 1 \right) \left( \frac{1}{2} - 2 \right)$ $= \left( \frac{1}{2} \right) \left( -\frac{1}{2} \right) \left( -\frac{3}{2} \right)$ $= \left( \frac{1}{2} \times -\frac{1}{2} \times -\frac{3}{2} \right) = \frac{3}{8} \quad (\text{positive})$

• Interval $(1, 2)$, choose $x = 1.5$:
  $(1.5)(1.5 - 1)(1.5 - 2) = (1.5)(0.5)(-0.5) = -0.375 \quad (\text{negative})$

• Interval $(2, \infty)$, choose $x = 3$:
  $(3)(3 - 1)(3 - 2) = (3)(2)(1) = 6 \quad (\text{positive})$

Step 3: Identify solution set

We need the regions where the expression is positive:

• $(0,1)$
• $(2, \infty)$

Since we have strict inequality $> 0$, we exclude the points where the expression is zero.

Final Answer:

$(0,1) \cup (2, \infty)$

---

Do you want more details or a graph to visualize this?

Related Questions:

1. How do we solve inequalities with more than three factors?
2. What happens if the inequality was $\geq 0$ instead?
3. How do we handle inequalities with negative coefficients?
4. Can we use sign charts to solve inequalities systematically?
5. How does the degree of the polynomial affect the number of sign changes?

Tip: Always test intervals by picking a number and substituting it into the expression to determine sign changes!