Let's solve the equation step by step:

The given equation is:

\[
6 - 2(x - 1) = -3(1 + x)
\]

### Step 1: Distribute the constants
Distribute \(-2\) on the left-hand side and \(-3\) on the right-hand side:

\[
6 - 2(x - 1) = 6 - 2x + 2
\]
\[
-3(1 + x) = -3 - 3x
\]

Now, the equation becomes:

\[
6 - 2x + 2 = -3 - 3x
\]

### Step 2: Combine like terms
On the left-hand side, combine \(6\) and \(2\):

\[
8 - 2x = -3 - 3x
\]

### Step 3: Move all \(x\) terms to one side and constants to the other
Add \(3x\) to both sides:

\[
8 - 2x + 3x = -3 - 3x + 3x
\]
\[
8 + x = -3
\]

Now, subtract 8 from both sides:

\[
x = -3 - 8
\]
\[
x = -11
\]

### Final Answer:
The solution to the equation is \(x = -11\).

Would you like any further details or explanations?

Here are 5 related questions you might want to explore:
1. How can you solve equations with variables on both sides?
2. What happens if there are fractions involved in such an equation?
3. Can you show how to solve equations involving exponents?
4. How can systems of equations be solved using substitution?
5. What are some techniques for solving quadratic equations?

**Tip**: Always check your solution by substituting the value of \(x\) back into the original equation!

Learn how to solve the linear equation 6-2(x-1) = -3(1+x) with a step-by-step solution. This problem covers key algebraic concepts like the distributive property and combining like terms, suitable for students in Grades 8-10.

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