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To solve the given linear equation:

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

### Step 1: Eliminate the fractions by finding a common denominator
The common denominator for \(4\), \(2\), and \(9\) is 36. Multiply every term by 36 to eliminate the fractions:

\[
36 \cdot \frac{x + 4}{4} = 36 \cdot \frac{3}{2} + 36 \cdot \frac{x - 6}{9}
\]

This simplifies to:

\[
9(x + 4) = 18 \cdot 3 + 4(x - 6)
\]

### Step 2: Distribute and simplify the equation
Now, distribute the constants:

\[
9x + 36 = 54 + 4x - 24
\]

Simplify further:

\[
9x + 36 = 4x + 30
\]

### Step 3: Solve for \(x\)
Subtract \(4x\) from both sides to combine like terms:

\[
5x + 36 = 30
\]

Subtract 36 from both sides:

\[
5x = -6
\]

Divide by 5:

\[
x = -\frac{6}{5}
\]

### Step 4: Check the solution
Substitute \(x = -\frac{6}{5}\) back into the original equation to verify:

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

Simplify each part:

1. \(-\frac{6}{5} + 4 = \frac{-6 + 20}{5} = \frac{14}{5}\)
2. \(\frac{14}{5} \div 4 = \frac{14}{5} \times \frac{1}{4} = \frac{14}{20} = \frac{7}{10}\)

For the right side:

1. \(-\frac{6}{5} - 6 = -\frac{6}{5} - \frac{30}{5} = -\frac{36}{5}\)
2. \(\frac{-\frac{36}{5}}{9} = -\frac{36}{5} \times \frac{1}{9} = -\frac{36}{45} = -\frac{4}{5}\)

The original equation becomes:

\[
\frac{7}{10} = \frac{3}{2} - \frac{4}{5}
\]

Find a common denominator:

\[
\frac{3}{2} - \frac{4}{5} = \frac{15}{10} - \frac{8}{10} = \frac{7}{10}
\]

Both sides are equal, so the solution \(x = -\frac{6}{5}\) is correct.

Would you like more details or have any questions? 

Here are 5 related questions you might consider:

1. What is the general method for solving linear equations with fractions?
2. How can you verify the solution of a linear equation?
3. What are some common mistakes to avoid when solving equations with fractions?
4. How can multiplying by a common denominator simplify solving equations?
5. How does one handle more complex linear equations involving multiple fractions?

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

Solve and check the linear equation \( \frac{x + 4}{4} = \frac{3}{2} + \frac{x - 6}{9} \).

Learn how to solve and check the linear equation \( \frac{x + 4}{4} = \frac{3}{2} + \frac{x - 6}{9} \) with a detailed step-by-step solution. This problem covers key algebraic concepts such as balancing equations, eliminating fractions, and verifying solutions. Suitable for students in Grades 8-10.

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