The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. A triangle has side lengths $4x + 20$, $2x + 8$, and $7x - 11$. We are tasked with determining the inequality that represents all possible values of $x$.

Solution:

Using the triangle inequality theorem, we write three inequalities:

1. $(4x + 20) + (2x + 8) > (7x - 11)$
2. $(2x + 8) + (7x - 11) > (4x + 20)$
3. $(7x - 11) + (4x + 20) > (2x + 8)$

1. Simplify the first inequality:

$$6x + 28 > 7x - 11$$ $$28 > x - 11$$ $$x < 39$$

2. Simplify the second inequality:

$$9x - 3 > 4x + 20$$ $$5x > 23$$ $$x > \frac{23}{5}$$

3. Simplify the third inequality:

$$11x + 9 > 2x + 8$$ $$9x > -1$$ $$x > -\frac{1}{9}$$

Since $x > -\frac{1}{9}$ is less restrictive and $x > \frac{23}{5}$ is more restrictive, we use $x > \frac{23}{5}$. Combine all inequalities:

$$\frac{23}{5} < x < 39$$

Multiply through by 5 to rewrite the inequality in terms of $5x$:

$$23 < 5x < 195$$

Conclusion: The inequality that represents all possible values of $x$ is:

Answer: D) $23 < 5x < 195$