Let's tackle each question one by one.

QUESTION 1

1.1

Given $T = \frac{\sqrt{x + 5}}{x - 1}$, where $x \in \{-6; -2; 1; 4\}$.

• 1.1.1 Rational: Find a value of $x$ that makes $T$ rational.
• 1.1.2 Non-real: Find a value of $x$ that makes $T$ non-real.

To answer these:

• For $T$ to be non-real, the square root $\sqrt{x + 5}$ must result in a negative number, meaning $x + 5 < 0$.
• For $T$ to be rational, $\sqrt{x + 5}$ should be a perfect square and $x - 1 \neq 0$.

1.2

Factorize the following expressions:

• 1.2.1 $3x - x^2$
• 1.2.2 $y - xy + x - 1$

Apply standard factorization techniques:

1. 1.2.1: Rewrite as $-x^2 + 3x = x(-x + 3)$.
2. 1.2.2: Factor by grouping.

1.3

Simplify the following:

• 1.3.1 $(x + 3)(x^2 - 3x + 9)$
• 1.3.2 $\frac{x^3 + x^2 + x + 1}{x^3 - 1} + \frac{2x + 2}{1 - x^2} + \frac{x + 1}{x^2 - 1}$
• 1.3.3 $\frac{10^{2x + 3} \cdot 4^{1 - x}}{25^{2x}}$

Each sub-question requires applying algebraic simplification, factorization, and properties of exponents.

1.4

Determine the numerical value of $(3x + y)^2$ given:

• $9x^2 + y^2 = 12$
• $xy = -3$

This can be solved by expressing $(3x + y)^2$ in terms of $x$ and $y$ and substituting the given equations.

QUESTION 2

2.1

Solve for $x$ without using a calculator:

• 2.1.1 $2x^2 - 5x = 3$
• 2.1.2 $\left(\frac{1}{2}\right)^{3x - 1} = 128$

For 2.1.1, rearrange and use the quadratic formula. For 2.1.2, express 128 as a power of $\frac{1}{2}$.

2.2

Solve for $x$ in the real numbers ($\mathbb{R}$):

• Inequality: $-5 \leq 3x - 2 < 7$

Solve by isolating $x$ within the bounds given.

2.3

Solve for $x$ and $y$ simultaneously:

• $2x - y = 17$
• $4y + 3x = 9$

This is a system of linear equations. Use substitution or elimination to find $x$ and $y$.

Would you like me to proceed with detailed solutions for each part, or is there a specific question you'd like to focus on?

Here are 5 follow-up questions that could extend this material:

1. How do we determine if a rational or non-real value is possible in expressions involving square roots?
2. What are alternative factorization techniques for polynomials with four terms?
3. How can exponent rules simplify complex expressions like those in 1.3.3?
4. What methods work best for solving inequalities involving multiple steps?
5. What are other methods for solving simultaneous equations besides substitution and elimination?

Tip: When solving complex expressions, keep track of each step and simplify as much as possible to avoid errors.