We are given the equation:

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

Step 1: Factor out $e^{-x}$

We notice that $e^{-x}$ is a common factor in all terms on the left-hand side. Let's factor it out:

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

Step 2: Simplify the constants inside the parentheses

Now let's compute the sum inside the parentheses:

$1.75 \cdot 0.5 = 0.875$ $1.75 \cdot 1.5 = 2.625$ $101.75 \cdot 2 = 203.5$

So, the equation becomes:

$e^{-x} \cdot \left( 0.875 + 1.75 + 2.625 + 203.5 \right) = 86.3995$

Simplifying the sum:

$0.875 + 1.75 + 2.625 + 203.5 = 208.75$

Thus, the equation reduces to:

$e^{-x} \cdot 208.75 = 86.3995$

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

To isolate $e^{-x}$, divide both sides by 208.75:

$e^{-x} = \frac{86.3995}{208.75}$

Calculating the right-hand side:

$e^{-x} \approx 0.4139$

Step 4: Solve for $x$

To solve for $x$, take the natural logarithm of both sides:

$-x = \ln(0.4139)$

$x = -\ln(0.4139)$

Now, let's compute $\ln(0.4139)$:

$\ln(0.4139) \approx -0.8827$

Thus:

$x \approx 0.8827$

Final Answer:

$x \approx 0.8827$

Would you like more details on any part of the solution?

Here are 5 related questions to deepen your understanding:

1. What is the general method for solving equations involving $e^x$?
2. How do you factor common terms in an exponential equation?
3. What is the significance of taking the natural logarithm to solve for $x$?
4. How can you solve a system of exponential equations?
5. What are some real-world applications of exponential decay and growth?

Tip: When solving exponential equations, always try to factor out common terms to simplify the equation before solving.