To solve this equation, $3 \sinh x - 2 \cosh x - 2 = 0$, we will use hyperbolic identities and solve for $x$ step-by-step.

Given:
$3 \sinh x - 2 \cosh x - 2 = 0$

Step 1: Rewrite using hyperbolic definitions

We know that: $\sinh x = \frac{e^x - e^{-x}}{2} \quad \text{and} \quad \cosh x = \frac{e^x + e^{-x}}{2}.$ Substitute these into the equation:

$3 \left(\frac{e^x - e^{-x}}{2}\right) - 2 \left(\frac{e^x + e^{-x}}{2}\right) - 2 = 0$

Step 2: Simplify the expression

Combine terms by multiplying through by 2 to eliminate the denominators:

$3(e^x - e^{-x}) - 2(e^x + e^{-x}) - 4 = 0$

Expanding each term gives:

$3e^x - 3e^{-x} - 2e^x - 2e^{-x} - 4 = 0$

Combine like terms:

[ (e^x - 5e^{-x}) - 4