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Let's go through each problem step-by-step.

### 1. Determine whether each statement is true
i. \( \mathbb{N} \subseteq \mathbb{V} \)
ii. \( \mathbb{Q} \subseteq \mathbb{R} \)
iii. \( \mathbb{R} \subseteq \mathbb{Q} \)

### Solution:
i. \( \mathbb{N} \subseteq \mathbb{V} \) (False, assuming \(\mathbb{V}\) denotes the set of vectors)
ii. \( \mathbb{Q} \subseteq \mathbb{R} \) (True)
iii. \( \mathbb{R} \subseteq \mathbb{Q} \) (False)

### 2. Express each of the following intervals as an inequality in the variable \( x \).
i. \((-4, 7)\)

### Solution:
i. \(-4 < x < 7\)

### 3. Simplify each of the following
i. \((-1.5, n) \cup (0.3, \infty)\)
ii. \(\{x: -2 < x \leq 3\} \cap \{x: x \neq 1 \text{ or } x \neq 2 \}\)

### Solution:
i. \((-1.5, n) \cup (0.3, \infty)\) (Assuming \( n \in \mathbb{R} \cup \{\infty\} \), simplify further if \( n = \infty \) gives \((-1.5, \infty)\))
ii. \(-2 < x \leq 3\), \( x \neq 1\), \( x \neq 2\)

### 4. Solve the equation \( (4x + 1)^2 = x + 3 \)

### Solution:
First, let \( y = 4x + 1 \), so the equation becomes:
\[ y^2 = x + 3 \]
\[ (4x + 1)^2 = x + 3 \]
\[ 16x^2 + 8x + 1 = x + 3 \]
\[ 16x^2 + 7x - 2 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-7 \pm \sqrt{49 + 128}}{32} \]
\[ x = \frac{-7 \pm \sqrt{177}}{32} \]

### 5. Simplify \( 5 \times 11^x \times 10^x = 22 \times 11^x \)

### Solution:
\[ 5 \times 11^x \times 10^x = 22 \times 11^x \]
\[ 5 \times 10^x = 22 \]
\[ 10^x = \frac{22}{5} \]
\[ x = \log_{10} \left(\frac{22}{5}\right) \]

### 6. Simplify the following
i. \( \sqrt[3]{27} \)

### Solution:
\[ \sqrt[3]{27} = 3 \]

### 7. Given \( \log_5 4 = 0.561 \) and \( \log_7 2 = 0.887 \), find each of the following:
i. \( \log_{35} 8 \)

### Solution:
\[ \log_{35} 8 = \frac{\log_{10} 8}{\log_{10} 35} \]
\[ \log_{10} 8 = 3 \log_{10} 2 \approx 3 \times 0.3010 = 0.9030 \]
\[ \log_{10} 35 = \log_{10} 5 + \log_{10} 7 \approx 0.6990 + 0.8451 = 1.5441 \]
\[ \log_{35} 8 \approx \frac{0.9030}{1.5441} \approx 0.585 \]

### 8. Simplify \( 2 + \log_3 8 \)

### Solution:
\[ 2 + \log_3 8 = 2 + \frac{\log_{10} 8}{\log_{10} 3} \]
\[ \log_{10} 8 \approx 0.9030 \]
\[ \log_{10} 3 \approx 0.4771 \]
\[ 2 + \frac{0.9030}{0.4771} \approx 2 + 1.893 \approx 3.893 \]

### 9. Simplify \( \log_4 (32) - \log_4 (2) \)

### Solution:
\[ \log_4 (32) - \log_4 (2) = \log_4 \left( \frac{32}{2} \right) = \log_4 (16) \]
\[ \log_4 (16) = 2 \]

### 10. Solve for \( u \) and \( v \) such that \( (m - 6u + 1)(v - 1) = 11 + 2 \)

### Solution:
\[ (m - 6u + 1)(v - 1) = 13 \]

Since there are multiple variables, more information or another equation is needed to uniquely determine \( u \) and \( v \).

If you have any questions or need further details, feel free to ask!

Here are 8 related questions for practice:
1. Determine if \( \mathbb{Z} \subseteq \mathbb{Q} \).
2. Express the interval \([2, 5)\) as an inequality.
3. Simplify the union \( (-2, 4) \cup (3, 7) \).
4. Solve the equation \( x^2 - 4x - 5 = 0 \).
5. Simplify \( 2 \times 3^x = 6 \).
6. Simplify \( \sqrt[4]{16} \).
7. Given \( \log_2 5 = 2.3219 \), find \( \log_2 25 \).
8. Simplify \( 3 + \log_2 9 \).

**Tip:** Always check the domain of the function while solving logarithmic and exponential equations to avoid undefined values.

Explore advanced problems in set theory, inequalities, equations, logarithms, and exponentials with detailed solutions. Practice using the quadratic formula and solving complex logarithmic identities. Suitable for advanced high school students and above.

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