https://file.aiquickdraw.com/mathbot/file/1726987735117wcf3gifv.jpg

Let's go through and solve the exponential inequalities from the image one by one:

### 6.) \( 3^{x-4} > 27 \)

1. Express 27 as a power of 3: 
   \[
   27 = 3^3
   \]
2. The inequality becomes:
   \[
   3^{x-4} > 3^3
   \]
3. Since the bases are the same, you can set the exponents:
   \[
   x - 4 > 3
   \]
4. Solve for \( x \):
   \[
   x > 7
   \]

### 7.) \( 2^{3x-4} < \frac{1}{32} \)

1. Express \(\frac{1}{32}\) as a power of 2:
   \[
   \frac{1}{32} = 2^{-5}
   \]
2. The inequality becomes:
   \[
   2^{3x-4} < 2^{-5}
   \]
3. Set the exponents:
   \[
   3x - 4 < -5
   \]
4. Solve for \( x \):
   \[
   3x < -1 \implies x < -\frac{1}{3}
   \]

### 8.) \( 2^x \geq 4x + 1 \)

This inequality mixes an exponential and a linear function, so solving it algebraically requires some thought. However, we can check for solutions by graphing both sides or using numerical methods.

For now, let's analyze it at some specific values of \(x\):

- For \(x = 0\):
  \[
  2^0 = 1 \quad \text{and} \quad 4(0) + 1 = 1
  \]
  So, \( 1 \geq 1 \), which is true.

- For \(x = 1\):
  \[
  2^1 = 2 \quad \text{and} \quad 4(1) + 1 = 5
  \]
  Here, \( 2 \geq 5 \) is false.

So, the solution seems to include \(x = 0\), and further analysis or graphing would show where the inequality holds.

### 9.) \( 4^{x-3} \leq \left(\frac{1}{8}\right)^{2x+1} \)

1. Express 4 and \( \frac{1}{8} \) as powers of 2:
   \[
   4 = 2^2 \quad \text{and} \quad \frac{1}{8} = 2^{-3}
   \]
2. The inequality becomes:
   \[
   \left(2^2\right)^{x-3} \leq \left(2^{-3}\right)^{2x+1}
   \]
3. Simplify the exponents:
   \[
   2^{2(x-3)} \leq 2^{-3(2x+1)}
   \]
4. Set the exponents:
   \[
   2(x - 3) \leq -3(2x + 1)
   \]
5. Simplify and solve for \( x \):
   \[
   2x - 6 \leq -6x - 3
   \]
   \[
   8x \leq 3
   \]
   \[
   x \leq \frac{3}{8}
   \]

### 10.) \( 25^{x+1} > 625^{x+2} \)

1. Express both 25 and 625 as powers of 5:
   \[
   25 = 5^2 \quad \text{and} \quad 625 = 5^4
   \]
2. The inequality becomes:
   \[
   \left(5^2\right)^{x+1} > \left(5^4\right)^{x+2}
   \]
3. Simplify the exponents:
   \[
   5^{2(x+1)} > 5^{4(x+2)}
   \]
4. Set the exponents:
   \[
   2(x+1) > 4(x+2)
   \]
5. Simplify and solve for \( x \):
   \[
   2x + 2 > 4x + 8
   \]
   \[
   -6 > 2x
   \]
   \[
   x < -3
   \]

---

Let me know if you would like further clarification on any of the solutions. Here are some related questions:

1. How can we solve inequalities involving logarithms?
2. What methods are useful for graphing exponential functions?
3. What is the difference between solving exponential inequalities and polynomial inequalities?
4. How can we convert between exponential and logarithmic forms to solve equations?
5. How do we check if an inequality solution satisfies the original equation?

**Tip:** Always verify solutions to inequalities by substituting values back into the original inequality or graphing both sides to check where they overlap.

Solve the exponential inequalities:
6.) 3^{x-4} > 27
7.) 2^{3x-4} < 1/32
8.) 2^x ≥ 4x + 1
9.) 4^{x-3} ≤ (1/8)^{2x+1}
10.) 25^{x+1} > 625^{x+2}

A detailed walkthrough on how to solve exponential inequalities such as 3^{x-4} > 27 and 2^{3x-4} < 1/32. This page covers key algebraic concepts, rules of exponents, and inequality solutions, suitable for high school students (Grades 10-12).

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