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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

Find the value of x if 3 sinh x - 2 cosh x - 2 = 0 using hyperbolic function formula.

Find the value of x in the hyperbolic equation 3 sinh x - 2 cosh x - 2 = 0 using hyperbolic function formulas. This detailed solution covers key hyperbolic function concepts, suitable for undergraduate students.

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