We need to solve the inequality:

\[
\left|(4x + 1) - 6\right| \geq 13
\]

### Step 1: Simplify the expression inside the absolute value
First, simplify the expression inside the absolute value:

\[
\left|4x + 1 - 6\right| \geq 13
\]

Simplify further:

\[
\left|4x - 5\right| \geq 13
\]

### Step 2: Set up two inequalities
The absolute value inequality \(\left|A\right| \geq B\) can be split into two cases:

1. \(4x - 5 \geq 13\)
2. \(4x - 5 \leq -13\)

### Step 3: Solve each inequality

#### Case 1: \(4x - 5 \geq 13\)
Add \(5\) to both sides:

\[
4x \geq 18
\]

Divide by \(4\):

\[
x \geq \frac{18}{4} = 4.5
\]

#### Case 2: \(4x - 5 \leq -13\)
Add \(5\) to both sides:

\[
4x \leq -8
\]

Divide by \(4\):

\[
x \leq \frac{-8}{4} = -2
\]

### Step 4: Write the solution
The solution to the inequality is:

\[
x \geq 4.5 \quad \text{or} \quad x \leq -2
\]

This can also be written in interval notation as:

\[
x \in (-\infty, -2] \cup [4.5, \infty)
\]

Would you like more details or have any questions?

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Here are 5 related questions:

1. How do you solve absolute value inequalities with "less than" instead of "greater than"?
2. What does the solution of an absolute value inequality look like on a number line?
3. Can you explain the difference between strict and non-strict inequalities?
4. How do absolute value inequalities change when the inequality is reversed?
5. What is the impact of negative coefficients inside the absolute value on the solution?

**Tip:** When solving absolute value inequalities, always split them into two separate cases to account for the positive and negative scenarios!

Understand how to solve the absolute value inequality |4x - 5| ≥ 13 with a clear step-by-step solution. This problem explores key algebraic concepts related to absolute value inequalities and is suitable for students in Grades 9-10.

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