Let's solve the equation step by step:

The equation is:

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

### Step 1: Distribute the constants.
Distribute the 4 and the -2 across the parentheses:

\[
4 \cdot 2x - 4 \cdot 3 - 2 \cdot x - 2 \cdot 4 = 2
\]

This simplifies to:

\[
8x - 12 - 2x - 8 = 2
\]

### Step 2: Combine like terms.
Combine the terms involving \(x\) and the constant terms:

\[
(8x - 2x) + (-12 - 8) = 2
\]

Simplifying further:

\[
6x - 20 = 2
\]

### Step 3: Isolate the variable.
Add 20 to both sides to isolate the \(x\) term:

\[
6x = 22
\]

### Step 4: Solve for \(x\).
Finally, divide both sides by 6:

\[
x = \frac{22}{6} = \frac{11}{3} \approx 3.67
\]

So, the solution is:

\[
x = \frac{11}{3}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the solution change if the equation was \(4(2x - 3) + 2(x + 4) = 2\)?
2. Can you solve \(3(4x - 5) - 2(2x + 7) = 6\) using the same steps?
3. What is the effect of changing the constant on the right side of the equation, say \(4(2x - 3) - 2(x + 4) = 10\)?
4. How would you solve the equation if it had a different form, like \(5x - 7 = 3(2x + 1)\)?
5. Can you solve an equation where the variable is in the denominator, like \(\frac{3x + 1}{2} - \frac{x - 4}{3} = 1\)?

**Tip:** When combining like terms, always pay attention to the signs (+ or -) to avoid common mistakes in simplification.

Learn how to solve the equation 4 times the quantity 2x minus 3, minus 2 times the quantity x plus 4, equals 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property and solving linear equations. Suitable for students in Grades 7-9.

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