Let's solve the equation \( 4(3x - 6) = 6(-4 + 2x) \).

### Step 1: Expand both sides

Expand both sides using the distributive property:

\[
4(3x - 6) = 4 \cdot 3x - 4 \cdot 6 = 12x - 24
\]
\[
6(-4 + 2x) = 6 \cdot (-4) + 6 \cdot 2x = -24 + 12x
\]

So, the equation becomes:

\[
12x - 24 = -24 + 12x
\]

### Step 2: Move all terms involving \(x\) to one side

Subtract \(12x\) from both sides:

\[
12x - 24 - 12x = -24 + 12x - 12x
\]

This simplifies to:

\[
-24 = -24
\]

### Step 3: Interpret the result

The equation \( -24 = -24 \) is true for all values of \(x\). Therefore, this is an identity, meaning that **the equation is true for all real numbers**.

The solution is:
\[
x \in \mathbb{R}
\]

Would you like further details on this solution or have any questions?

Here are 5 related questions you might find interesting:
1. What is an identity equation, and how does it differ from a conditional equation?
2. How do you solve equations where no solution exists?
3. Can you solve an equation like \( 5x + 3 = 5x + 7 \)?
4. What happens if you encounter an equation with fractions, such as \( \frac{3x}{2} = 4 \)?
5. How can you check if an equation has infinite solutions, like the one above?

**Tip:** Always expand terms first when solving equations with parentheses to simplify the process.

Learn how to solve the identity equation 4(3x - 6) = 6(-4 + 2x) using the distributive property. This solution covers key algebraic concepts and explains why this equation holds true for all real numbers. Ideal for students in Grades 8-10.

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