These are exponential inequalities. I'll walk through each one step-by-step to solve them.

1. $5^{x+2} > 25$
   Rewrite 25 as $5^2$:
   $5^{x+2} > 5^2$
   Since the bases are the same, compare the exponents:
   $x + 2 > 2$
   $x > 0$

2. $9^x < 81$
   Rewrite 81 as $9^2$:
   $9^x < 9^2$
   $x < 2$

3. $2^{x-1} < 4^3$
   Rewrite $4$ as $2^2$, so $4^3 = (2^2)^3 = 2^6$:
   $2^{x-1} < 2^6$
   $x - 1 < 6$
   $x < 7$

4. $7^{4x+3} \geq 7^{x-6}$
   Since the bases are the same, compare the exponents:
   $4x + 3 \geq x - 6$
   $3x \geq -9$
   $x \geq -3$

5. $3^{6x+2} \geq 9^{2x}$
   Rewrite $9$ as $3^2$, so $9^{2x} = (3^2)^{2x} = 3^{4x}$:
   $3^{6x+2} \geq 3^{4x}$
   $6x + 2 \geq 4x$
   $2x \geq -2$
   $x \geq -1$

6. $\left(\frac{1}{2}\right)^{3x+4} < \left(\frac{1}{4}\right)^{2x-1}$
   Rewrite $\frac{1}{4}$ as $(\frac{1}{2})^2$:
   $\left(\frac{1}{2}\right)^{3x+4} < \left(\frac{1}{2}\right)^{2(2x-1)}$
   $\left(\frac{1}{2}\right)^{3x+4} < \left(\frac{1}{2}\right)^{4x-2}$
   Since the bases are the same and are fractions, the inequality sign flips when comparing exponents:
   $3x + 4 > 4x - 2$
   $6 > x$
   $x < 6$

7. $\left(\frac{1}{9}\right)^{2x-1} \geq \left(\frac{1}{27}\right)^x$
   Rewrite $\frac{1}{9}$ as $(\frac{1}{3})^2$ and $\frac{1}{27}$ as $(\frac{1}{3})^3$:
   $\left(\frac{1}{3}\right)^{2(2x-1)} \geq \left(\frac{1}{3}\right)^{3x}$
   $\left(\frac{1}{3}\right)^{4x-2} \geq \left(\frac{1}{3}\right)^{3x}$
   Since the bases are the same, compare the exponents:
   $4x - 2 \geq 3x$
   $x \geq 2$

8. $\left(\frac{1}{8}\right)^{2x+1} \geq \left(\frac{1}{4}\right)^{x+2}$
   Rewrite $\frac{1}{8}$ as $(\frac{1}{2})^3$ and $\frac{1}{4}$ as $(\frac{1}{2})^2$:
   $\left(\frac{1}{2}\right)^{3(2x+1)} \geq \left(\frac{1}{2}\right)^{2(x+2)}$
   $\left(\frac{1}{2}\right)^{6x+3} \geq \left(\frac{1}{2}\right)^{2x+4}$
   Compare the exponents:
   $6x + 3 \leq 2x + 4$
   $4x \leq 1$
   $x \leq \frac{1}{4}$

Let me know if you'd like further details or have any questions on any step!

Here are five related questions to expand on this topic:

1. How would you solve exponential inequalities with different bases that cannot be expressed as powers of each other?
2. What happens to the inequality sign when dealing with bases between 0 and 1?
3. How do logarithmic functions help solve more complex exponential inequalities?
4. Can you solve inequalities involving irrational bases, such as $e^x$?
5. What are some real-world applications of exponential inequalities?

Tip: Remember, when you have an inequality with a fractional base, reversing the inequality sign is necessary when comparing exponents, as fractional powers have an inverse relationship with magnitude.