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Let's go through each equation and solve for \( x \).

### 1) \( 16^{2x - 3} = 4^{x + 2} \)

Express both sides with base 2:

\[
16 = 2^4 \quad \text{and} \quad 4 = 2^2
\]

Thus, the equation becomes:

\[
(2^4)^{2x - 3} = (2^2)^{x + 2}
\]

Simplify:

\[
2^{4(2x - 3)} = 2^{2(x + 2)}
\]

Equating exponents:

\[
4(2x - 3) = 2(x + 2)
\]

Solve for \( x \):

\[
8x - 12 = 2x + 4
\]

\[
6x = 16
\]

\[
x = \frac{8}{3}
\]

---

### 2) \( \left(\frac{1}{2}\right)^{2x} = 2^{3 - x} \)

Rewrite \( \frac{1}{2} \) as \( 2^{-1} \):

\[
(2^{-1})^{2x} = 2^{3 - x}
\]

Simplify:

\[
2^{-2x} = 2^{3 - x}
\]

Equating exponents:

\[
-2x = 3 - x
\]

Solve for \( x \):

\[
-2x + x = 3
\]

\[
-x = 3 \quad \Rightarrow \quad x = -3
\]

---

### 3) \( 4^{2x+7} \leq 32^{2x-3} \)

Express both sides with base 2:

\[
4 = 2^2 \quad \text{and} \quad 32 = 2^5
\]

Rewrite the equation:

\[
(2^2)^{2x + 7} \leq (2^5)^{2x - 3}
\]

Simplify:

\[
2^{2(2x + 7)} \leq 2^{5(2x - 3)}
\]

Equating exponents:

\[
4x + 14 \leq 10x - 15
\]

Solve for \( x \):

\[
14 + 15 \leq 10x - 4x
\]

\[
29 \leq 6x
\]

\[
x \geq \frac{29}{6}
\]

---

### 4) \( \left(\frac{2}{5}\right)^{5x - 1} \geq \frac{25}{4} \)

Rewrite \( \frac{25}{4} \) as \( \left(\frac{5}{2}\right)^2 \):

\[
\left(\frac{2}{5}\right)^{5x - 1} \geq \left(\frac{5}{2}\right)^2
\]

Taking logarithms (or using exponents properties):

\[
5x - 1 \leq -2
\]

Solve for \( x \):

\[
5x \leq -1
\]

\[
x \leq -\frac{1}{5}
\]

---

### 5) \( \left(\frac{2}{3}\right)^{5x + 2} = \left(\frac{3}{2}\right)^{2x} \)

Take the logarithm of both sides:

\[
(5x + 2) \log\left(\frac{2}{3}\right) = 2x \log\left(\frac{3}{2}\right)
\]

Note that \( \log\left(\frac{2}{3}\right) = -\log\left(\frac{3}{2}\right) \). The equation becomes:

\[
-(5x + 2) = 2x
\]

Solve for \( x \):

\[
-5x - 2 = 2x
\]

\[
-7x = 2
\]

\[
x = -\frac{2}{7}
\]

---

Would you like me to continue with the rest of the questions, or do you need further clarification on these solutions? Here are some related questions you might be interested in:

1. How do logarithmic properties help solve exponential equations?
2. Can all exponential equations be solved by converting to the same base?
3. What are the general steps to solving inequalities involving exponents?
4. How do you handle fractional bases in exponential equations?
5. Why do exponential functions often appear in growth or decay models?

**Tip**: When solving equations involving exponents, converting both sides to the same base often simplifies the process significantly!

Solve for x in the given set of exponential equations.

Step-by-step solutions to solve exponential equations, involving key algebraic and logarithmic concepts. Suitable for high school students, covering exponent properties, solving inequalities, and converting to logarithmic forms.

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