To solve the inequality:
$x^2 - 4x \leq 0$

Step 1: Factor the inequality

Factor the left side: $x(x - 4) \leq 0$

Step 2: Find the critical points

Set each factor equal to zero: $x = 0 \quad \text{or} \quad x = 4$ These are the boundary points where the expression equals zero.

Step 3: Test intervals

Divide the real number line into three intervals:
$(- \infty, 0), (0, 4), (4, \infty)$.

We test each interval by selecting a point within the interval:

1. For $x = -1$ in $(- \infty, 0)$:
   $(-1)(-1 - 4) = (-1)(-5) = 5 > 0$
   (This interval does not satisfy the inequality.)

2. For $x = 2$ in $(0, 4)$:
   $(2)(2 - 4) = (2)(-2) = -4 < 0$
   (This interval satisfies the inequality.)

3. For $x = 5$ in $(4, \infty)$:
   $(5)(5 - 4) = (5)(1) = 5 > 0$
   (This interval does not satisfy the inequality.)

Step 4: Include boundary points

Since the inequality is $\leq 0$, the boundary points where the expression equals zero are included. These are $x = 0$ and $x = 4$.

Final Answer:

$[0, 4]$

Would you like further clarification or graphing of the solution?

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Related Questions:

1. How would the solution change if the inequality were $x^2 - 4x < 0$?
2. What is the graphical representation of this inequality?
3. How do we solve similar quadratic inequalities systematically?
4. Can we use the quadratic formula here instead of factoring?
5. How does the sign test help in solving inequalities like this?

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Tip: Always remember to test intervals between critical points for inequalities involving polynomials.