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Let's solve the given problem step by step. We have two parts to address:

1. Solve the equation symbolically:
   \[
\frac{5}{8^x} - \frac{1}{5} = -\frac{1}{5}
   \]

2. Solve the related inequality:
   \[
\frac{5}{8^x} - \frac{1}{5} \leq -\frac{1}{5}
   \]

### Step 1: Solve the Equation
\[
\frac{5}{8^x} - \frac{1}{5} = -\frac{1}{5}
\]
Let's add \(\frac{1}{5}\) to both sides:
\[
\frac{5}{8^x} = 0
\]

For \(\frac{5}{8^x}\) to equal 0, \(8^x\) would need to be infinite. However, since \(x\) is real, \(8^x\) is always positive, and \(\frac{5}{8^x}\) cannot equal 0. Thus, **there is no solution** for the equation.

### Step 2: Solve the Inequality
\[
\frac{5}{8^x} - \frac{1}{5} \leq -\frac{1}{5}
\]
Add \(\frac{1}{5}\) to both sides:
\[
\frac{5}{8^x} \leq 0
\]

Similar to the equation, \(\frac{5}{8^x}\) is always positive for any real value of \(x\) since \(8^x > 0\) for all real \(x\). Thus, there is no value of \(x\) that satisfies this inequality.

### Conclusion:
The correct answer is:

**B. There is no solution.**

Would you like further details or explanations on this? Here are some related questions to deepen your understanding:

1. What happens to \(8^x\) as \(x\) increases or decreases?
2. Why can't \(\frac{5}{8^x}\) be negative?
3. How would the solution change if the equation were \(\frac{5}{8^x} + \frac{1}{5} = \frac{1}{5}\)?
4. What is the general approach to solving exponential equations?
5. How does adding or subtracting terms affect inequalities?

**Tip:** When dealing with exponential expressions, remember that \(a^x\) is always positive for positive \(a\), which affects the possible solutions of equations and inequalities.

Solve the equation symbolically. Then solve the related inequality: (5 / 8^x) - (1/5) = - (1/5). Solve (5 / 8^x) - (1/5) ≤ - (1/5).

Learn how to solve the exponential equation (5 / 8^x) - (1/5) = - (1/5) and its related inequality (5 / 8^x) - (1/5) ≤ - (1/5). The solution involves understanding properties of exponents and inequality principles, suitable for students in Grades 10-12.

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