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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?

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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.

The user submitted a set of mathematical problems from the image, including equations to solve, domain and range questions, function analysis, graph-related problems, and more.

Explore how to solve various mathematical problems involving logarithms, quadratic equations, and functions. The solution covers domain and range, one-to-one and onto functions, and graphing functions. Suitable for students in Grades 10-12.

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