https://file.aiquickdraw.com/mathbot/file/1725442516301tazegnlu.jpg

The given equation from the image is:

\[
\frac{2}{3} \left[ 2 - \left( \frac{x}{3} + 6 \right) - 2(x - 1) \right] = \frac{5}{6} x - 1
\]

Let's solve this equation step-by-step.

### Step 1: Simplify the Expression Inside the Brackets

First, let's simplify the expression inside the brackets:

\[
2 - \left( \frac{x}{3} + 6 \right) - 2(x - 1)
\]

Distribute the \(-2\) in the second term:

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

Combine like terms:

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

### Step 2: Multiply by \(\frac{2}{3}\)

Now multiply the simplified expression by \(\frac{2}{3}\):

\[
\frac{2}{3} \left( -2 - \frac{x}{3} - 2x \right)
\]

Distribute \(\frac{2}{3}\):

\[
\frac{2}{3} \times (-2) + \frac{2}{3} \times \left( -\frac{x}{3} \right) + \frac{2}{3} \times (-2x)
\]

This simplifies to:

\[
-\frac{4}{3} - \frac{2x}{9} - \frac{4x}{3}
\]

### Step 3: Combine Terms on Both Sides

Now, the equation becomes:

\[
-\frac{4}{3} - \frac{2x}{9} - \frac{4x}{3} = \frac{5}{6}x - 1
\]

### Step 4: Clear the Denominators

To eliminate the fractions, multiply the entire equation by 18 (the least common multiple of 3, 6, and 9):

\[
18 \left( -\frac{4}{3} - \frac{2x}{9} - \frac{4x}{3} \right) = 18 \left( \frac{5}{6}x - 1 \right)
\]

This simplifies to:

\[
-24 - 4x - 24x = 15x - 18
\]

### Step 5: Combine Like Terms

Combine like terms:

\[
-28x - 24 = 15x - 18
\]

Add \(28x\) to both sides:

\[
-24 = 43x - 18
\]

Add 18 to both sides:

\[
-6 = 43x
\]

### Step 6: Solve for \(x\)

Divide both sides by 43:

\[
x = -\frac{6}{43}
\]

### Solution

The solution for \(x\) is:

\[
\boxed{x = -\frac{6}{43}}
\]

If you have any questions or need further details, feel free to ask!

### Related Questions:

1. How would the solution change if there were additional terms inside the brackets?
2. What if the equation had a different fraction multiplier on the left-hand side?
3. How can we verify the solution by substituting \(x = -\frac{6}{43}\) back into the original equation?
4. What methods can be used to solve similar equations with fractions?
5. What is the geometric interpretation of the given equation?

**Tip:** Always check your work by substituting the solution back into the original equation to verify it is correct.

Solve the equation: (2/3)[2 - (x/3 + 6) - 2(x - 1)] = (5/6)x - 1

Learn how to solve the fractional algebraic equation (2/3)[2 - (x/3 + 6) - 2(x - 1)] = (5/6)x - 1 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property and combining like terms. Suitable for students in Grades 9-10.

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