To solve the inequality and match the solution with the provided answer choices, let's revisit the solution step-by-step and determine the correct interval.

Given Inequality:

$\left( \frac{1}{5} \right)^{\frac{2x + 1}{1 - x}} > \left( \frac{1}{5} \right)^{-3}$

Simplifying the Inequality:

Since the base $\frac{1}{5}$ is less than 1, the inequality flips when dealing with exponents: $\frac{2x + 1}{1 - x} < -3$

Solving the Exponent Inequality:

1. Start with: $\frac{2x + 1}{1 - x} < -3$

2. Cross-multiply: $2x + 1 < -3(1 - x)$

3. Distribute $-3$ on the right side: $2x + 1 < -3 + 3x$

4. Isolate $x$: $2x + 1 < -3 + 3x$ Subtract $2x$ from both sides: $1 < -3 + x$ Add 3 to both sides: $4 < x \quad \text{or} \quad x > 4$

This means the solution to the inequality is $x > 4$.

Matching with the Given Choices:

Let's check which interval matches $x > 4$:

• Option a: $x \in (-\infty, 1) \cup (4, +\infty)$ -- This option includes $x > 4$.
• Option b: $x \in (-1, 1)$ -- This option does not include $x > 4$.
• Option c: $x \in (1, 4)$ -- This option does not include $x > 4$.
• Option d: $x \in (-\infty, -1) \cup (1, +\infty)$ -- This option includes $x > 4$.

The correct choice is option a, $x \in (-\infty, 1) \cup (4, +\infty)$, as it includes the solution $x > 4$.

Would you like any further details or have any questions?

Related Questions:

1. How does the base of an exponential expression affect the direction of an inequality?
2. What is the process for solving inequalities involving fractions?
3. How can you simplify inequalities with negative exponents?
4. How do you handle inequalities when the base is between 0 and 1?
5. Can you provide a similar inequality problem to solve?
6. How would the inequality change if the base were greater than 1?
7. What steps should be taken if the inequality had different bases?
8. How can inequalities involving exponents be graphed?

Tip:

Always verify the intervals provided in multiple-choice questions by comparing them to the solution set derived from solving the inequality. This helps to ensure the correct choice is selected.