https://file.aiquickdraw.com/mathbot/file/1728262934015q3mhmtwq.jpg

We are tasked with determining all positive numbers \( A \) for which two conditions hold.

### Step-by-step Solution:

#### 1. Analyze the first condition:
\[
\text{If } |x - 2| < 1 \text{ then } |4x - 8| < A
\]

First, simplify \( |4x - 8| \):
\[
|4x - 8| = 4|x - 2|
\]
Therefore, the condition becomes:
\[
|x - 2| < 1 \implies 4|x - 2| < A
\]
Since \( |x - 2| < 1 \), multiplying both sides by 4 gives:
\[
4|x - 2| < 4
\]
Thus, we need \( A > 4 \) for the first inequality to hold.

#### 2. Analyze the second condition:
\[
\text{If } |x + 5| < 4 \text{ then } |-3x - 15| < A
\]

Simplify \( |-3x - 15| \):
\[
|-3x - 15| = 3|x + 5|
\]
Therefore, the condition becomes:
\[
|x + 5| < 4 \implies 3|x + 5| < A
\]
Since \( |x + 5| < 4 \), multiplying both sides by 3 gives:
\[
3|x + 5| < 12
\]
Thus, we need \( A > 12 \) for the second inequality to hold.

### Conclusion:
For both inequalities to hold, \( A \) must be greater than 12. Therefore:
\[
A > 12
\]

Would you like a deeper explanation on any step or have any questions? Here are some related questions to think about:

1. What is the geometric interpretation of absolute value inequalities?
2. How does multiplying an absolute value inequality by a constant affect the solution?
3. How would the solution change if we had a different range for \( x \)?
4. What happens if we introduce additional constraints on \( A \)?
5. Can we generalize this method to other similar inequalities?

**Tip:** When dealing with absolute value inequalities, always consider breaking the inequality into two cases or simplifying using properties like \( |ax + b| = a|x + b/a| \).

Determine all numbers A > 0 for which the following statements hold:
1. If |x - 2| < 1 then |4x - 8| < A.
2. If |x + 5| < 4 then |-3x - 15| < A.

Explore how to solve inequalities involving absolute values and linear transformations in this detailed breakdown. Learn key algebraic concepts, such as simplifying absolute value expressions and applying properties of inequalities. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation