To solve for \( w \) in the equation
\[ 4 + \frac{3}{4}w = 2 + \frac{2}{3}w, \]
let's start by isolating the variable \( w \).
First, we can move the terms involving \( w \) to one side and the constant terms to the other side. We will subtract \( \frac{2}{3}w \) from both sides:
\[ 4 + \frac{3}{4}w - \frac{2}{3}w = 2. \]
Next, we can also subtract 4 from both sides:
\[ \frac{3}{4}w - \frac{2}{3}w = 2 - 4, \]
which simplifies to
\[ \frac{3}{4}w - \frac{2}{3}w = -2. \]
Now to combine the terms on the left, we need a common denominator for \( \frac{3}{4} \) and \( \frac{2}{3} \). The least common multiple of 4 and 3 is 12.
Rewriting the fractions:
\[ \frac{3}{4}w = \frac{9}{12}w, \] \[ \frac{2}{3}w = \frac{8}{12}w. \]
Now we substitute these back into our equation:
\[ \frac{9}{12}w - \frac{8}{12}w = -2. \]
Subtracting the fractions gives:
\[ \frac{1}{12}w = -2. \]
Now, to solve for \( w \), we multiply both sides by 12:
\[ w = -2 \cdot 12, \]
which results in:
\[ w = -24. \]
Thus, the solution is
\[ \boxed{-24}. \]