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The document contains several mathematical problems. Here’s a breakdown of each:

---

### Problem 1:
You are asked to calculate the values of two variables, $ a $ and $ b $, given the following:

#### Expressions for $ a $ and $ b $:
\[
a = \frac{1}{8} \cdot \log_2(16) + \log_c(c^3) + \log_7\left(\sqrt{7^{49}}\right) \quad (c > 0, c \neq 1)
\]
\[
b = \sum_{k=11}^{13} \left(10 - \frac{k}{2}\right)
\]

##### Sub-questions:
1. Calculate $ a $ and $ b $.
2. Using the values of $ a $ and $ b $, solve the following inequalities:
   - $ \left| 12x - 1 \right| > a $
   - $ \left| 12x - 1 \right| \leq b $

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### Problem 2:
A supermarket offers two promotions for purchasing 6 packages of pasta, each costing $1250:

1. **40% off on the second unit.**
2. **Buy 3 and pay for 2.**

You are asked to:
- Determine which promotion offers the greatest savings.
- Calculate how much you save by choosing the most advantageous promotion.

---

### Problem 3:
Given a linear function $ f : \mathbb{R} \to \mathbb{R} $ whose graph passes through the points $ (1, -3) $ and $ (-1, 9) $:

1. Find a formula for $ f(x) $ and solve for $ x \in \mathbb{R} $ where $ f(x) < 0 $, writing the solution in interval notation.
2. Find the equation of a line perpendicular to the graph of $ f $ that passes through the point $ (6, 4) $, and graph both lines.

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Would you like to start solving any specific problem? Let me know if you need explanations or help with calculations!

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**Five Related Questions**:
1. How do you calculate the logarithmic expression for different bases?
2. How do you solve absolute value inequalities?
3. How do promotions like "buy one, get one 40% off" work mathematically?
4. What is the formula to find the equation of a line passing through two points?
5. How can you find the equation of a line perpendicular to another?

**Tip**: When solving inequalities involving absolute values, split them into two cases: positive and negative scenarios for the expression inside the absolute value.

Considerar a ∈ R y de b ∈ R siendo :

a = \frac{1}{8} \cdot \log_2(16) + \log_c(c^3) + \log_7(\sqrt{7^{49}}) \quad (c > 0, c \neq 1)
b = \sum_{k=11}^{13} \left(10 - \frac{k}{2}\right)

Calcular a y b. Considerar los valores de a y de b hallados en el ítem anterior. Hallar el conjunto solución de:
1. |12x - 1| > a
2. |12x - 1| ≤ b

Learn how to calculate variables a and b using logarithmic expressions and summation, and solve absolute value inequalities. This example includes solving logarithmic equations and inequalities, with detailed solutions for undergraduate students.

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