Let's go through the problems one by one.

1. Solve the following equation using suitable methods:

a. $\dfrac{2}{x} - \dfrac{x}{x^2 - 2} = \dfrac{1}{2} - \dfrac{2x - 2}{3x - 2}$

b. $\dfrac{1}{1 + \dfrac{1}{x}} + \dfrac{1}{1 - \dfrac{1}{x}} = \dfrac{x + 1}{x - 1}$

c. $\log 2 + \log (x + 2) - \log (3x - 5) = \log 3$

d. $2x^2 - x - 2 < 0$

2. Find the domain and range of the following functions:

a. $f(x) = 1 + 8x - 2x^2$

b. $f(x) = \dfrac{1}{x^2 - 5x + 6}$

c. $f(x) = \sqrt{x^2 - 6x + 8}$

3. Determine whether the following functions are one-to-one, onto, or neither:

a. $f : \mathbb{R} \to \mathbb{R}, \ f(x) = |x|$

b. $f : \mathbb{R} \to \mathbb{R}, \ f(x) = x^2$

4. Find $b$ and $c$ given that the graph of $y = x^2 + bx + c$ passes through the points $(3, 14)$ and $(-4, 7)$.

5. Show that $3 + \sqrt{5}$ is irrational and simplify expressions involving logarithms.

6. Let $f(x) = 14x - 23$, $g(x) = \dfrac{1}{x^2}$, $h(x) = x^2 - x$. Find $(f \circ g)(x + 1)$ and evaluate $f(20), g(2), h(10)$.

7. Draw the graph of the functions:

a. $f(x) = x^4 - 3x^3 - 4x^2 + x + 1$

b. $f(x) = \dfrac{3x^2}{(x - 2)(x + 1)}$

Would you like to start solving these problems step by step or focus on a particular one?

Here are 5 questions to explore further:

1. How would you solve a quadratic inequality like in question 1d?
2. What is the graphical interpretation of the domain and range of a function?
3. How can we check if a function is one-to-one and/or onto?
4. What strategies help in solving logarithmic equations like the one in question 1c?
5. How do we verify if a number is irrational?

Tip: When solving inequalities, always check for critical points by solving the equation as if it were equal, then test intervals between those critical points.