The document contains a series of math problems covering topics such as inequalities, logarithms, trigonometry, matrices, partial fractions, and more. Here's a breakdown of some of the problems in the document:

1. Inequality: Solve $\frac{1}{x} + \frac{1}{x+2} < \frac{1}{x+1}$.
2. Graphical method: Solve the inequality $|x - 1| > \frac{1}{2}x + 1$ graphically.
3. Logarithmic equation: Given $\log_a n = x$ and $\log_b n = y$, show the relationship between $x$ and $y$.
4. Trigonometric identity: Prove $\sin A \cdot \sin(B - C) + \sin B \cdot \sin(C - A) + \sin C \cdot \sin(A - B) = 0$.
5. Partial Fractions: Decompose $\frac{x^3 + 4x^2 - 10x + 6}{(x - 3)(x^2 + 4)}$ into partial fractions.

These questions range from algebraic manipulation to solving inequalities and working with logarithms, matrices, and trigonometric identities. If you'd like, I can walk you through the solutions for any of these problems.

Do you want details or have any specific questions? Here are 5 related follow-up questions:

1. How do you solve inequalities involving fractions algebraically?
2. What are the steps to solving an equation graphically?
3. How can logarithmic properties be applied to solve for variables in equations?
4. What trigonometric identities are commonly used in proofs?
5. How do you approach problems involving matrix operations?

Tip: When solving trigonometric identities, look for opportunities to apply sum and difference formulas for simplification.