Let's solve the given exponential inequalities one by one.

Problem 1: Solve $3^{x+1} > 243$

Solution:

1. Write 243 as a power of 3:
   $243 = 3^5$

2. Rewrite the inequality:
   $3^{x+1} > 3^5$

3. Since the bases are the same and the exponential function is strictly increasing, we can compare the exponents:
   $x + 1 > 5$

4. Solve for $x$:
   $x > 4$

Problem 2: Solve $25^2 \geq 5^{x+1}$

Solution:

1. Calculate $25^2$:
   $25 = 5^2 \implies 25^2 = (5^2)^2 = 5^4$

2. Rewrite the inequality:
   $5^4 \geq 5^{x+1}$

3. Since the bases are the same, compare the exponents:
   $4 \geq x + 1$

4. Solve for $x$:
   $4 - 1 \geq x \implies x \leq 3$

Problem 3: Solve $\left(\frac{49}{81}\right)^{3x+1} \geq \frac{9}{7}$

Solution:

1. Rewrite the terms as powers of 7 and 9:
   $\frac{49}{81} = \left(\frac{7}{9}\right)^2$
   $\frac{9}{7} = \left(\frac{7}{9}\right)^{-1}$

2. Rewrite the inequality:
   $\left(\left(\frac{7}{9}\right)^2\right)^{3x+1} \geq \left(\frac{7}{9}\right)^{-1}$

3. Simplify the left side:
   $\left(\frac{7}{9}\right)^{2(3x+1)} \geq \left(\frac{7}{9}\right)^{-1}$

4. Since $\frac{7}{9} < 1$, reverse the inequality when comparing exponents:
   $2(3x+1) \leq -1$

5. Solve for $x$:
   $6x + 2 \leq -1$
   $6x \leq -3$
   $x \leq -\frac{1}{2}$

Problem 4: Solve $\left(\frac{1}{9}\right)^{3x+5} \leq \left(\frac{1}{243}\right)^{x-6}$

Solution:

1. Rewrite the bases as powers of 3:
   $\frac{1}{9} = 3^{-2}$
   $\frac{1}{243} = 3^{-5}$

2. Rewrite the inequality:
   $(3^{-2})^{3x+5} \leq (3^{-5})^{x-6}$

3. Simplify the exponents:
   $3^{-2(3x+5)} \leq 3^{-5(x-6)}$

4. Since the bases are the same and both sides have negative exponents (exponential function is decreasing), we can compare the exponents:
   $-2(3x+5) \geq -5(x-6)$

5. Solve for $x$:
   $-6x - 10 \geq -5x + 30$
   $-x \geq 40$
   $x \leq -40$

Summary of Solutions:

1. $x > 4$
2. $x \leq 3$
3. $x \leq -\frac{1}{2}$
4. $x \leq -40$

Would you like more details or explanations for any of these steps?

Related Questions:

1. How do you solve exponential equations involving different bases?
2. What are some common techniques for solving inequalities?
3. How can you convert a fraction to a power of a base?
4. What happens to an inequality when you multiply or divide by a negative number?
5. What is the importance of recognizing increasing and decreasing functions in inequalities?

Tip:

Always pay attention to the base of the exponent. When the base is between 0 and 1, the direction of the inequality changes when you compare the exponents.