https://file.aiquickdraw.com/mathbot/file/1726472476678nwuyoyzb.jpg

You've provided a mathematical inequality and implication from a book: 

\[ |x + 4| < \frac{\epsilon}{2} \Rightarrow |12x + 8| < \epsilon \]

Let's analyze and validate this implication:

1. **Understanding Absolute Values:**
   - The absolute value \( |x + 4| < \frac{\epsilon}{2} \) implies that the distance of \(x + 4\) from zero is less than \(\frac{\epsilon}{2}\). This can be rewritten as:
     \[
     -\frac{\epsilon}{2} < x + 4 < \frac{\epsilon}{2}
     \]
   - By isolating \(x\), we get:
     \[
     -\frac{\epsilon}{2} - 4 < x < \frac{\epsilon}{2} - 4
     \]

2. **Analyzing the Implication:**
   - To see if this implies \( |12x + 8| < \epsilon \), we substitute the bounds of \(x\) into \(12x + 8\):
     \[
     12\left(-\frac{\epsilon}{2} - 4\right) + 8 \quad \text{and} \quad 12\left(\frac{\epsilon}{2} - 4\right) + 8
     \]
   - Simplifying these expressions:
     \[
     12\left(-\frac{\epsilon}{2} - 4\right) + 8 = -6\epsilon - 48 + 8 = -6\epsilon - 40
     \]
     \[
     12\left(\frac{\epsilon}{2} - 4\right) + 8 = 6\epsilon - 48 + 8 = 6\epsilon - 40
     \]
   - This results in:
     \[
     -6\epsilon - 40 < 12x + 8 < 6\epsilon - 40
     \]

3. **Final Validation:**
   - We need \( |12x + 8| < \epsilon \), which translates to:
     \[
     -\epsilon < 12x + 8 < \epsilon
     \]
   - Given our bounds of \(12x + 8\) as \([-6\epsilon - 40, 6\epsilon - 40]\), the range does not immediately guarantee that it is within \((-\epsilon, \epsilon)\) for all values of \(\epsilon\). The additional \(-40\) term in the derived range can potentially cause it to exceed the desired bounds of \(-\epsilon\) to \(\epsilon\), unless further conditions on \(\epsilon\) or \(x\) are specified.

Therefore, the given implication \( |x + 4| < \frac{\epsilon}{2} \Rightarrow |12x + 8| < \epsilon \) is not universally true without additional restrictions on \(\epsilon\) or \(x\).

Would you like more details on this topic or have any questions? Here are some related questions you might find useful:

1. How do the terms in the inequality affect the range of solutions?
2. What additional conditions on \(\epsilon\) or \(x\) could make the implication always true?
3. Can this type of implication be generalized for other coefficients or constants?
4. How does scaling the variable \(x\) in the absolute value affect the inequality's solution?
5. What are the implications of altering the epsilon value in terms of real-world applications?

**Tip:** When working with inequalities and absolute values, always consider the effect of any constants or coefficients on the bounds of your solution.

Solve inequalities involving absolute values and find delta (δ) in terms of epsilon (ε).

Learn how to solve absolute value inequalities and find delta (δ) in terms of epsilon (ε) using epsilon-delta analysis. This problem involves understanding the relationship between bounds on x and corresponding limits on functions, a key concept in calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation