Let's solve the equation step-by-step:

\[ \frac{4}{7}x - x = -\frac{9}{14} \]

First, let's combine the terms involving \( x \):

\[ \frac{4}{7}x - x = \frac{4}{7}x - \frac{7}{7}x = \left(\frac{4}{7} - \frac{7}{7}\right)x = \left(\frac{4}{7} - 1\right)x = \left(\frac{4}{7} - \frac{7}{7}\right)x = \left(\frac{4 - 7}{7}\right)x = -\frac{3}{7}x \]

So the equation becomes:

\[ -\frac{3}{7}x = -\frac{9}{14} \]

Next, we can eliminate the negative signs on both sides:

\[ \frac{3}{7}x = \frac{9}{14} \]

To solve for \( x \), we need to isolate \( x \). We can do this by multiplying both sides of the equation by the reciprocal of \( \frac{3}{7} \), which is \( \frac{7}{3} \):

\[ x = \frac{9}{14} \times \frac{7}{3} \]

Now, multiply the fractions:

\[ x = \frac{9 \cdot 7}{14 \cdot 3} = \frac{63}{42} \]

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 21:

\[ x = \frac{63 \div 21}{42 \div 21} = \frac{3}{2} \]

So, the solution to the equation is:

\[ x = \frac{3}{2} \]