Let's solve the equation:

$1.75e^{-0.5x} + 1.75e^{-x} + 1.75e^{-1.5x} + 101.75e^{-2x} = 86.3995$

Step 1: Simplify the equation

First, let's factor out $1.75$ from the first three terms:

$1.75 \left( e^{-0.5x} + e^{-x} + e^{-1.5x} \right) + 101.75e^{-2x} = 86.3995$

Next, divide the entire equation by 1.75 to further simplify:

$e^{-0.5x} + e^{-x} + e^{-1.5x} + 58.14e^{-2x} = 49.37114$

Step 2: Apply substitution

Let $y = e^{-0.5x}$, so:

$y^2 = e^{-x}, \quad y^3 = e^{-1.5x}, \quad y^4 = e^{-2x}$

Substitute into the equation:

$y + y^2 + y^3 + 58.14y^4 = 49.37114$

Step 3: Solve the polynomial equation

This is now a polynomial equation in $y$:

$58.14y^4 + y^3 + y^2 + y - 49.37114 = 0$

To find the value of $y$, this equation needs to be solved using numerical methods, such as Newton's method or by using a computational tool. Once $y$ is found, solve for $x$ using $y = e^{-0.5x}$.

Let's solve this numerically to find the value of $y$.The numerical solution for $y$ is approximately $y \approx 0.9466$.

Step 4: Solve for $x$

Recall that $y = e^{-0.5x}$. So,

$0.9466 = e^{-0.5x}$

Take the natural logarithm on both sides:

$\ln(0.9466) = -0.5x$

Solving for $x$:

$x = -2 \ln(0.9466)$

Now, let's compute the value of $x$.The solution for $x$ is approximately $x \approx 0.1098$.

Would you like any further details or have any questions? Here are some related problems you might find interesting:

1. Solve a similar equation with different coefficients.
2. Explore the solution method for a non-exponential polynomial equation.
3. Analyze the behavior of the equation as one of the coefficients increases.
4. Consider the case where the constant term on the right-hand side changes.
5. Investigate the solutions for higher-order exponential equations.

Tip: When dealing with equations involving exponential terms, substituting variables can simplify the problem into a polynomial equation, making it easier to solve.