To find $\tanh(x)$, given the equation:

$6 \sinh(x) + 2 \cosh(x) + 7 = 0$

Step 1: Express the hyperbolic functions in terms of exponentials.

We know that:

• $\sinh(x) = \frac{e^x - e^{-x}}{2}$
• $\cosh(x) = \frac{e^x + e^{-x}}{2}$

Substituting these into the equation:

$6 \left(\frac{e^x - e^{-x}}{2}\right) + 2 \left(\frac{e^x + e^{-x}}{2}\right) + 7 = 0$

Step 2: Simplify the equation.

Simplifying the terms:

$3(e^x - e^{-x}) + (e^x + e^{-x}) + 7 = 0$

Combine like terms:

$(3e^x + e^x) - (3e^{-x} - e^{-x}) + 7 = 0$

This simplifies to:

$4e^x - 2e^{-x} + 7 = 0$

Step 3: Solve for $e^x$ or $e^{-x}$.

Let's solve this equation for $e^x$.

1. Move the constant to the other side:

$4e^x - 2e^{-x} = -7$

This is a transcendental equation and typically requires solving through a substitution or numerical method. We can proceed by assuming a substitution or numerical approach to find a precise solution. Let's solve it numerically to determine the value of $x$ and then find $\tanh(x)$.

Let me calculate this for you.The solutions for $x$ are complex expressions involving logarithms. However, the values of $\tanh(x)$ for the solutions are:

1. $\tanh\left(\log\left(\frac{7}{2} - \frac{\sqrt{37}}{2}\right)\right)$
2. $\tanh\left(\log\left(\frac{\sqrt{37}}{2} + \frac{7}{2}\right)\right)$

These expressions represent the hyperbolic tangent of $x$ for the two solutions. They can be further evaluated numerically if needed.

Would you like the numerical values of $\tanh(x)$, or do you need further clarification?

Here are five related questions for deeper understanding:

1. How are hyperbolic functions related to exponential functions?
2. What is the geometric interpretation of $\tanh(x)$?
3. How can transcendental equations be solved numerically?
4. What are the differences between trigonometric and hyperbolic functions?
5. Can we use a graphical method to estimate solutions to hyperbolic equations?

Tip: Hyperbolic functions are analogs of trigonometric functions, but they are based on exponential functions rather than circular functions.