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To solve the inequality \( |4x - 1| \leq 7 \) and write the solution in interval notation, we need to break down the absolute value inequality into two cases.

For an absolute value inequality of the form \( |A| \leq B \), where \( B \geq 0 \), we can rewrite this as:
\[
-B \leq A \leq B
\]

### Step 1: Set up the inequality
So, for \( |4x - 1| \leq 7 \), we rewrite this as:
\[
-7 \leq 4x - 1 \leq 7
\]

### Step 2: Solve the compound inequality
Now, we’ll solve this inequality in two parts.

1. **First part**: \(-7 \leq 4x - 1\)

   Add 1 to both sides:
   \[
   -7 + 1 \leq 4x \implies -6 \leq 4x
   \]
   Now, divide by 4:
   \[
   -\frac{6}{4} \leq x \implies -\frac{3}{2} \leq x
   \]

2. **Second part**: \(4x - 1 \leq 7\)

   Add 1 to both sides:
   \[
   4x \leq 7 + 1 \implies 4x \leq 8
   \]
   Now, divide by 4:
   \[
   x \leq \frac{8}{4} \implies x \leq 2
   \]

### Step 3: Combine the results
We have found:
\[
-\frac{3}{2} \leq x \leq 2
\]

### Step 4: Write the solution in interval notation
In interval notation, this solution is:
\[
\left[ -\frac{3}{2}, 2 \right]
\]

### Answer
So, the solution set is:
**A. The solution set is** \(\left[ -\frac{3}{2}, 2 \right]\).

Would you like further details on any of the steps, or do you have any questions?

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Here are some related questions you might find useful:

1. How do we solve absolute value inequalities in general?
2. What happens if the inequality sign changes to \(>| |4x - 1| > 7\)?
3. Can we use interval notation for inequalities involving infinity?
4. How can we graph the solution set on a number line?
5. What is the difference between solving an absolute inequality with \(\leq\) and \(\geq\)?

**Tip:** When solving absolute value inequalities, remember that \(|A| \leq B\) implies \( -B \leq A \leq B \).

Solve the inequality |4x - 1| ≤ 7 and write the solution in interval notation.

Learn how to solve the absolute value inequality |4x - 1| ≤ 7 and write the solution in interval notation. This problem covers key algebraic concepts, including absolute value inequalities and interval notation, suitable for Grades 9-11.

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