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Solve the following absolute value inequality. Write your answer in interval notation. Must show work to receive FULL credit!

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!

Solve the following absolute value inequality: |4x - 1| ≤ 27. Write your answer in interval notation.

Learn how to solve the absolute value inequality |4x - 1| ≤ 27 with a detailed, step-by-step solution, including interval notation. This problem covers key algebraic concepts related to absolute values and inequalities. Suitable for students in Grades 8-10.

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