To solve the given equation, let's first simplify the expression step by step. The equation is:

\[
\frac{X - \frac{x}{12} + \frac{x}{18}}{7 - \frac{7}{12} + \frac{7}{18}} = -\frac{4}{7}
\]

### Step 1: Simplify the Numerator

The numerator is:

\[
X - \frac{x}{12} + \frac{x}{18}
\]

To combine the terms, we need a common denominator. The least common multiple of 12 and 18 is 36. So, we rewrite each fraction with a denominator of 36:

\[
X - \frac{x}{12} + \frac{x}{18} = X - \frac{3x}{36} + \frac{2x}{36} = X - \frac{3x - 2x}{36} = X - \frac{x}{36}
\]

So, the simplified numerator is:

\[
X - \frac{x}{36}
\]

### Step 2: Simplify the Denominator

The denominator is:

\[
7 - \frac{7}{12} + \frac{7}{18}
\]

Again, we need a common denominator. The least common multiple of 12 and 18 is 36. So, we rewrite each fraction with a denominator of 36:

\[
7 - \frac{7}{12} + \frac{7}{18} = 7 - \frac{21}{36} + \frac{14}{36} = 7 - \frac{21 - 14}{36} = 7 - \frac{7}{36}
\]

So, the simplified denominator is:

\[
7 - \frac{7}{36}
\]

### Step 3: Substitute the Simplified Expressions Back into the Equation

Now we substitute the simplified numerator and denominator back into the equation:

\[
\frac{X - \frac{x}{36}}{7 - \frac{7}{36}} = -\frac{4}{7}
\]

### Step 4: Simplify the Denominator Further

The denominator \(7 - \frac{7}{36}\) can be written as:

\[
7 - \frac{7}{36} = \frac{252}{36} - \frac{7}{36} = \frac{252 - 7}{36} = \frac{245}{36}
\]

So, the equation becomes:

\[
\frac{X - \frac{x}{36}}{\frac{245}{36}} = -\frac{4}{7}
\]

### Step 5: Simplify the Entire Fraction

To simplify the left-hand side, multiply both the numerator and the denominator by 36:

\[
\frac{36 \left(X - \frac{x}{36}\right)}{245} = -\frac{4}{7}
\]

This simplifies to:

\[
\frac{36X - x}{245} = -\frac{4}{7}
\]

### Step 6: Cross-Multiply to Solve for \(X\)

Cross-multiply to solve for \(X\):

\[
7(36X - x) = -4 \times 245
\]

Simplify both sides:

\[
252X - 7x = -980
\]

### Step 7: Solve for \(X\)

Combine like terms:

\[
252X - 7x = -980
\]

Factor out the common term \(x\):

\[
x(252 - 7) = -980
\]

Simplify the coefficient:

\[
x \cdot 245 = -980
\]

Finally, solve for \(x\):

\[
x = \frac{-980}{245} = -4
\]

So, the solution is:

\[
x = -4
\]