Let's solve the inequality step by step.

Given inequality:

$-6(x + 1) < 2(x + 5)$

Step 1: Distribute the terms on both sides.

On the left side, distribute $-6$ across $(x + 1)$, and on the right side, distribute $2$ across $(x + 5)$:

$-6x - 6 < 2x + 10$

Step 2: Combine like terms.

Now, move all the $x$-terms to one side and the constant terms to the other side. First, add $6x$ to both sides:

$-6 < 8x + 10$

Then, subtract $10$ from both sides:

$-16 < 8x$

Step 3: Solve for $x$.

Now, divide both sides by 8:

$\frac{-16}{8} < x$

$-2 < x$

This means that:

$x > -2$

Final Answer:

The solution to the inequality is:

$x > -2$

Do you want more details on any step, or have any questions?

Here are 5 related questions for you:

1. How would the solution change if the inequality were $-6(x + 1) \leq 2(x + 5)$?
2. What happens if you multiply or divide both sides of an inequality by a negative number?
3. Can you graph the solution $x > -2$ on a number line?
4. How do you check if a particular value of $x$ satisfies the inequality?
5. How would the inequality change if the signs inside the parentheses were reversed?

Tip: Always remember to reverse the inequality sign when multiplying or dividing both sides by a negative number!