We are given the inequality:

$3(x - 4) \geq 5x + 2$

We need to solve for $x$.

Step-by-step Solution:

1. Expand the left side: $3(x - 4) = 3x - 12$ The inequality becomes: $3x - 12 \geq 5x + 2$

2. Move all terms involving $x$ to one side: Subtract $3x$ from both sides: $-12 \geq 2x + 2$

3. Isolate the $x$-term: Subtract 2 from both sides: $-14 \geq 2x$

4. Solve for $x$: Divide both sides by 2: $-7 \geq x \quad \text{or} \quad x \leq -7$

The solution is $x \leq -7$. Now, let's check which of the given options satisfies this condition:

• $-10 \leq -7$ (True)
• $-5 \leq -7$ (False)
• $5 \leq -7$ (False)
• $10 \leq -7$ (False)

The correct answer is $\boxed{-10}$.

Do you have any questions or would you like more details? Here are five related questions to further explore:

1. What happens if we change the inequality from $\geq$ to $>$?
2. How would the solution change if we had a quadratic inequality instead of a linear one?
3. Can inequalities always be solved using the same process as equations?
4. What is the impact of multiplying or dividing by a negative number in inequalities?
5. How can we check the solution graphically?

Tip: Always double-check your work when solving inequalities, especially when dealing with negative numbers or reversing inequality signs!