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Let's work through each of the problems one by one. We need to show that the indicated implications are true.

### Problem 49:
We are given:
\[
|x - 3| < 0.5 \implies |5x - 15| < 2.5
\]

**Step 1: Solve the left-hand inequality \( |x - 3| < 0.5 \)**

This inequality means the distance between \(x\) and 3 is less than 0.5, so:
\[
-0.5 < x - 3 < 0.5
\]
Adding 3 to all parts:
\[
2.5 < x < 3.5
\]

So, \(x\) is bounded between 2.5 and 3.5.

**Step 2: Verify the right-hand inequality \( |5x - 15| < 2.5 \)**

We rewrite \( |5x - 15| \) as \( 5|x - 3| \) because:
\[
5x - 15 = 5(x - 3)
\]
Thus, the inequality becomes:
\[
5|x - 3| < 2.5
\]
Now divide by 5:
\[
|x - 3| < 0.5
\]

This is exactly the inequality we were given in the left-hand side of the implication. Therefore, the implication is true.

### Problem 50:
We are given:
\[
|x + 2| < 0.3 \implies |4x + 8| < 1.2
\]

**Step 1: Solve the left-hand inequality \( |x + 2| < 0.3 \)**

This inequality means the distance between \(x\) and -2 is less than 0.3, so:
\[
-0.3 < x + 2 < 0.3
\]
Subtracting 2 from all parts:
\[
-2.3 < x < -1.7
\]

So, \(x\) is bounded between -2.3 and -1.7.

**Step 2: Verify the right-hand inequality \( |4x + 8| < 1.2 \)**

We rewrite \( |4x + 8| \) as \( 4|x + 2| \) because:
\[
4x + 8 = 4(x + 2)
\]
Thus, the inequality becomes:
\[
4|x + 2| < 1.2
\]
Now divide by 4:
\[
|x + 2| < 0.3
\]

This is exactly the inequality we were given in the left-hand side of the implication. Therefore, the implication is true.

### Conclusion:
Both implications are true.

Do you want any further details on these problems or additional clarifications?

### Related Questions:
1. What are the steps for solving absolute value inequalities?
2. How can we rewrite expressions to simplify inequalities like these?
3. How does dividing or multiplying inequalities affect the solution?
4. What is the geometric interpretation of an absolute value inequality?
5. Can you give an example of when an implication like these might not be true?

### Tip:
When dealing with absolute value inequalities, always try to express the inequality as a standard range, and then verify if any transformations lead to equivalent inequalities.

Do number 49-50: Solve the implications involving absolute value inequalities.

Solve absolute value inequalities and verify implications. Learn how to work through Problems 49-50, where we analyze |x - 3| < 0.5 and |x + 2| < 0.3 to verify their respective implications with |5x - 15| < 2.5 and |4x + 8| < 1.2.

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