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!

Step-by-step solution to the linear inequality -6(x + 1) < 2(x + 5). Learn how to distribute terms, combine like terms, and solve for x using algebraic techniques. This guide is suitable for students in Grades 8-10.

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