Let's solve the equation step by step:

Given:

\[
5x - 2(4 + x) = 5 + 3(1 + x)
\]

### Step 1: Distribute the numbers outside the parentheses.
Distribute \(-2\) on the left side and \(3\) on the right side:

\[
5x - 2 \cdot 4 - 2 \cdot x = 5 + 3 \cdot 1 + 3 \cdot x
\]

This simplifies to:

\[
5x - 8 - 2x = 5 + 3 + 3x
\]

### Step 2: Combine like terms.
On the left side:

\[
(5x - 2x) - 8 = 3x - 8
\]

On the right side:

\[
5 + 3 = 8
\]

So the equation becomes:

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

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

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

This simplifies to:

\[
-8 = 8
\]

### Step 4: Solve for the variable.
At this point, we have \(-8 = 8\), which is a **contradiction**. This means there is **no solution** to this equation. The equation is **inconsistent**.

### Final Answer:
There is **no solution** for the equation \(5x - 2(4 + x) = 5 + 3(1 + x)\).

---

Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How can we identify if an equation has no solution?
2. What is the difference between a consistent and an inconsistent equation?
3. Can you provide an example of an equation that has infinitely many solutions?
4. How would the equation change if one side were modified slightly?
5. How does distributing numbers in an equation affect the solution process?

**Tip:** Always check for contradictions after simplifying equations, as they can indicate no solutions.

This page provides a detailed step-by-step solution to the linear equation 5x - 2(4 + x) = 5 + 3(1 + x). Learn how to apply the distributive property, combine like terms, and identify inconsistent equations. Suitable for students in Grades 8-10.

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