\frac{1}{6}\left(\frac{x}{2}-2\right)-3\left(\frac{x}{4}-1\right)

To solve the expression \(\frac{1}{6}\left(\frac{x}{2}-2\right)-3\left(\frac{x}{4}-1\right)\), we can simplify it step-by-step.

1. Distribute the \(\frac{1}{6}\) and \(-3\) to their respective terms:

   \[
   = \frac{1}{6} \cdot \frac{x}{2} - \frac{1}{6} \cdot 2 - 3 \cdot \frac{x}{4} + 3 \cdot 1
   \]

   \[
   = \frac{x}{12} - \frac{2}{6} - \frac{3x}{4} + 3
   \]

   \[
   = \frac{x}{12} - \frac{1}{3} - \frac{3x}{4} + 3
   \]

2. Convert \(-\frac{1}{3}\) and \(3\) to have a common denominator (12):

   \[
   -\frac{1}{3} = -\frac{4}{12}
   \]
   
   \[
   3 = \frac{36}{12}
   \]

3. Rewrite the expression:

   \[
   = \frac{x}{12} - \frac{4}{12} - \frac{9x}{12} + \frac{36}{12}
   \]

4. Combine like terms:

   \[
   = \frac{x - 9x - 4 + 36}{12}
   \]

   \[
   = \frac{-8x + 32}{12}
   \]

5. Simplify:

   \[
   = \frac{-2x + 8}{3}
   \]

Thus, the simplified expression is:

\[
\frac{-2x + 8}{3}
\]

Would you like to explore further details or have any questions? Here are some related questions you might find interesting:

1. How can we solve for \(x\) if the expression equals zero?
2. What would happen if we change the coefficients in the original expression?
3. How do we find the value of the expression for a specific \(x\)?
4. Can you explain how to simplify expressions with different types of functions?
5. What is the importance of common denominators in algebraic expressions?

**Tip:** Always check if you can factor the final expression for easier interpretation or further calculations!

Discover how to simplify the expression \frac{1}{6}\left(\frac{x}{2}-2\right)-3\left(\frac{x}{4}-1\right\) step-by-step. This solution involves key algebraic concepts such as the distributive property and finding common denominators, suitable for Grades 7-9.

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