To solve the absolute value inequality:

$|4x - 1| \leq 27$

Step 1: Break down the inequality

The absolute value inequality $|A| \leq B$ can be rewritten as a compound inequality:

$- B \leq A \leq B$

In this case, $A = 4x - 1$ and $B = 27$, so we rewrite the inequality as:

$-27 \leq 4x - 1 \leq 27$

Step 2: Solve the compound inequality

Now, we solve the inequality in two parts.

Part 1: Solve $-27 \leq 4x - 1$

Add 1 to both sides:

$-27 + 1 \leq 4x$ $-26 \leq 4x$

Now, divide both sides by 4:

$\frac{-26}{4} \leq x$ $x \geq -6.5$

Part 2: Solve $4x - 1 \leq 27$

Add 1 to both sides:

$4x \leq 27 + 1$ $4x \leq 28$

Now, divide both sides by 4:

$x \leq 7$

Step 3: Combine the results

We now have:

$-6.5 \leq x \leq 7$

Step 4: Write the solution in interval notation

The solution in interval notation is:

$[-6.5, 7]$

This is the set of all $x$ values between -6.5 and 7, inclusive.

Would you like any further details or clarification?

Here are 5 related questions to expand your understanding:

1. How would the solution change if the inequality were $|4x - 1| < 27$?
2. What is the difference between $\leq$ and $<$ when solving absolute value inequalities?
3. Can you solve the inequality $|4x - 1| \geq 27$ and describe the interval notation?
4. How does solving absolute value inequalities differ from solving regular linear inequalities?
5. How would you approach solving $|4x - 1| \leq k$ where $k$ is an unknown constant?

Tip: Always remember to check your final interval solution by testing boundary points to ensure they satisfy the original inequality!