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 $|ax + b| = a|x + b/a|$.