To find the truth set of the predicate $P(x): x^2 \leq 4x$, we will solve the inequality $x^2 \leq 4x$ for integer values of $x$.

Step 1: Rewrite the inequality

We start by rewriting the inequality: $x^2 \leq 4x$ Subtract $4x$ from both sides: $x^2 - 4x \leq 0$ Factor the left-hand side: $x(x - 4) \leq 0$

Step 2: Solve the inequality $x(x - 4) \leq 0$

To solve the inequality $x(x - 4) \leq 0$, we determine the critical points by setting each factor equal to 0: $x = 0 \quad \text{or} \quad x - 4 = 0 \quad \Rightarrow \quad x = 4$ These are the points where the expression changes sign.

Step 3: Test intervals around the critical points

The critical points divide the real line into three intervals:

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

Now, test the sign of $x(x - 4)$ in each interval:

• For $x \in (-\infty, 0)$ (e.g., $x = -1$): $(-1)(-1 - 4) = (-1)(-5) = 5 > 0$ The product is positive, so $P(x)$ is false in this interval.

• For $x \in (0, 4)$ (e.g., $x = 2$): $2(2 - 4) = 2(-2) = -4 < 0$ The product is negative, so $P(x)$ is true in this interval.

• For $x \in (4, \infty)$ (e.g., $x = 5$): $5(5 - 4) = 5(1) = 5 > 0$ The product is positive, so $P(x)$ is false in this interval.

Step 4: Check the critical points $x = 0$ and $x = 4$

• At $x = 0$: $0(0 - 4) = 0$ So, $P(0)$ is true.

• At $x = 4$: $4(4 - 4) = 0$ So, $P(4)$ is true.

Step 5: Conclusion

The truth set of the predicate $P(x): x^2 \leq 4x$ is the set of integers $x$ for which $x(x - 4) \leq 0$, i.e., $\boxed{\{ x \in \mathbb{Z} \mid 0 \leq x \leq 4 \}}.$ In interval notation, the truth set is $\{0, 1, 2, 3, 4\}$.

Would you like further details or have any questions?

Here are five related questions:

1. How would the truth set change if the domain were all real numbers instead of integers?
2. How can we determine the solution graphically?
3. What happens if we modify the predicate to $P(x): x^2 < 4x$?
4. How would the inequality change if the coefficient of $x$ were different?
5. Can this method be applied to solve other quadratic inequalities?

Tip: When solving quadratic inequalities, always factor the expression and check the sign of the product over different intervals.