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.