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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.

Solve the following exponential inequalities: 1) 3^(x+1) > 243, 2) 25^2 ≥ 5^(x+1), 3) (49/81)^(3x+1) ≥ 9/7, 4) (1/9)^(3x+5) ≤ (1/243)^(x-6)

Explore step-by-step solutions to exponential inequalities such as 3^(x+1) > 243 and (1/9)^(3x+5) ≤ (1/243)^(x-6). Understand exponent properties and learn how to compare powers with different bases. Suitable for Grades 10-12.

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