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### Zadanie 1: Rozwiąż równania

#### a)
\[
\frac{4x}{x^2} = 2 - \frac{x - 2}{x - 3}
\]

To solve:
1. Combine the fractions on the right-hand side.
2. Multiply through by the denominators to eliminate fractions.
3. Solve the resulting equation for \(x\).

#### b)
\[
\frac{x + 2}{x - 1} = \frac{3x}{x + 3}
\]

To solve:
1. Cross-multiply to get rid of the denominators.
2. Simplify the resulting equation.
3. Solve for \(x\).

---

### Zadanie 2: Rozwiąż nierówności

#### a)
\[
\frac{-4x - 3}{x - 1} < 4x
\]

To solve:
1. Move all terms to one side of the inequality.
2. Find common denominators.
3. Solve for \(x\), considering the sign change when multiplying or dividing by negative numbers.

#### b)
\[
\frac{x - 1}{x + 6} \geq 2
\]

To solve:
1. Subtract 2 from both sides.
2. Combine the expression into a single fraction.
3. Solve the inequality, taking care to check the domains of the rational function.

---

### Zadanie 3: Problem z książką

In this problem, you are given a total number of verses (8400) and asked how many verses are on each page and how many pages the book has if changing the number of verses per page reduces the total number of pages by 30.

Let \( p \) be the number of pages and \( v \) the number of verses per page. Initially:
\[
p \cdot v = 8400
\]

When 5 fewer verses per page are printed, the number of pages increases by 30. This gives a second equation:
\[
(p + 30) \cdot (v - 5) = 8400
\]

Solve this system of equations to find \( p \) and \( v \).

---

### Zadanie 4: Wykaż, że

You are asked to prove the following identity:
\[
\frac{a + b}{a + b + 0} = \frac{a + b}{a + b} + \frac{0}{a + b}
\]
and
\[
\frac{a + b}{\frac{a}{2} + \frac{a}{2} + b} = \frac{a}{a + b} = 1
\]

To prove these, break down each term and show equality step by step using basic algebra.

---

### Zadanie 5: Dana jest funkcja \( f(x) = \frac{|x + 1|}{x + 2} \)

The task is to draw the graph of this function and find the values of the parameter \( m \) such that the equation:
\[
\frac{|x + 1|}{x + 2} = m
\]
has exactly 2 solutions.

1. Analyze the behavior of the function \( f(x) \) based on the absolute value.
2. Graph the function considering the points where \( x + 1 = 0 \) and \( x + 2 = 0 \).
3. Identify the values of \( m \) that result in exactly two solutions.

---

Would you like to see detailed solutions for one or more of these problems?

### 5 Related Questions:
1. How do you solve inequalities involving rational expressions?
2. What is the impact of a negative number in inequality solving steps?
3. How can systems of equations be applied to real-world problems like this book problem?
4. What are key points to remember when solving absolute value equations?
5. How does cross-multiplication work in rational equations?

#### Tip: Always check the domain of rational functions to avoid division by zero errors.

Learn how to solve various algebra problems, including rational equations, inequalities, and absolute value functions. This includes a word problem involving systems of equations. Suitable for Grades 10-12 students.

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