Let's solve the equation step by step.

The given equation is:

\[
\frac{1}{6} \left(\frac{3}{2}j - 4\right) + j = \frac{2}{3}j
\]

### Step 1: Distribute \(\frac{1}{6}\) across the terms inside the parentheses:

\[
\frac{1}{6} \times \frac{3}{2}j - \frac{1}{6} \times 4 + j = \frac{2}{3}j
\]

This simplifies to:

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

Further simplification gives:

\[
\frac{1}{4}j - \frac{2}{3} + j = \frac{2}{3}j
\]

### Step 2: Combine the like terms on the left side:

Convert \(j\) into a fraction with a denominator of 4 to combine it with \(\frac{1}{4}j\):

\[
\frac{1}{4}j + \frac{4}{4}j - \frac{2}{3} = \frac{2}{3}j
\]

\[
\frac{5}{4}j - \frac{2}{3} = \frac{2}{3}j
\]

### Step 3: Isolate the variable \(j\):

Move the term \(\frac{2}{3}j\) from the right side to the left side:

\[
\frac{5}{4}j - \frac{2}{3}j = \frac{2}{3}
\]

Find a common denominator to combine the \(j\) terms:

\[
\frac{15}{12}j - \frac{8}{12}j = \frac{2}{3}
\]

This simplifies to:

\[
\frac{7}{12}j = \frac{2}{3}
\]

### Step 4: Solve for \(j\):

Multiply both sides by the reciprocal of \(\frac{7}{12}\):

\[
j = \frac{\frac{2}{3}}{\frac{7}{12}} = \frac{2}{3} \times \frac{12}{7} = \frac{24}{21} = \frac{8}{7}
\]

So, the solution is:

\[
j = \frac{8}{7}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you simplify fractions with different denominators?
2. What are some methods to isolate variables in linear equations?
3. Can you explain how to distribute a fraction over an expression inside parentheses?
4. What is the process of finding a common denominator?
5. How do you solve equations involving complex fractions?
6. How do you interpret solutions of equations with fractional coefficients?
7. Can you explain how to check your solution after solving an equation?
8. What are the common pitfalls when solving equations with fractions?

**Tip:** Always double-check your work by substituting the solution back into the original equation to ensure it satisfies the equation.

Learn how to solve the linear equation (1/6)(3/2j - 4) + j = 2/3j with a detailed step-by-step solution. This problem covers linear equations and fraction operations, suitable for students in Grades 7-9.

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