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To solve the inequality \( |5 - 7x| \leq 12 \), we’ll break it down into two cases due to the absolute value.

The inequality \( |5 - 7x| \leq 12 \) means:
\[
-12 \leq 5 - 7x \leq 12
\]

Now, we solve these two inequalities separately.

### Step 1: Solve \( -12 \leq 5 - 7x \)
1. Subtract 5 from both sides:
   \[
   -17 \leq -7x
   \]
2. Divide by \(-7\) and reverse the inequality sign:
   \[
   x \leq \frac{17}{7}
   \]

### Step 2: Solve \( 5 - 7x \leq 12 \)
1. Subtract 5 from both sides:
   \[
   -7x \leq 7
   \]
2. Divide by \(-7\) and reverse the inequality sign:
   \[
   x \geq -1
   \]

### Combine the Solutions
The solution to \( -12 \leq 5 - 7x \leq 12 \) is:
\[
-1 \leq x \leq \frac{17}{7}
\]

### Interval Notation
The solution in interval notation is:
\[
\left[-1, \frac{17}{7}\right]
\]

### Answer
The correct answer is:
- **A.** The solution set is \( \left[-1, \frac{17}{7}\right] \).

Would you like further details or explanations? 

Here are five related questions to explore further:
1. How would the solution change if the inequality were \( |5 - 7x| < 12 \)?
2. How do you solve absolute value inequalities when the inequality sign is "greater than" instead of "less than"?
3. What happens to the solution if the inequality becomes \( |5 - 7x| \geq 12 \)?
4. How do you interpret the solution interval \( \left[-1, \frac{17}{7}\right] \) on a number line?
5. Can you explain why we reverse the inequality when dividing by a negative number?

**Tip:** Always remember to flip the inequality sign when dividing or multiplying both sides by a negative value in inequalities.

Solve the inequality |5 - 7x| ≤ 12. Write the solution in interval notation.

Learn how to solve the absolute value inequality |5 - 7x| ≤ 12 and express the solution in interval notation. This problem covers essential concepts in algebra, including absolute value inequalities and interval notation. Suitable for high school students in Grades 9-11.

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