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?

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 $-B \leq A \leq B$.