Let's solve the inequality:

\[
|6u| - 5 \geq -1
\]

### Step 1: Add 5 to both sides of the inequality to isolate the absolute value term.
\[
|6u| - 5 + 5 \geq -1 + 5
\]
\[
|6u| \geq 4
\]

### Step 2: Solve the absolute value inequality.
Since \( |6u| \geq 4 \), we can break this into two cases:

1. \( 6u \geq 4 \)
2. \( 6u \leq -4 \)

### Step 3: Solve each case.

**Case 1:**  
\[
6u \geq 4
\]
Divide both sides by 6:
\[
u \geq \frac{4}{6} = \frac{2}{3}
\]

**Case 2:**  
\[
6u \leq -4
\]
Divide both sides by 6:
\[
u \leq \frac{-4}{6} = \frac{-2}{3}
\]

### Step 4: Combine the results.
The solution is:
\[
u \geq \frac{2}{3} \quad \text{or} \quad u \leq \frac{-2}{3}
\]

In interval notation, the solution is:
\[
u \in (-\infty, -\frac{2}{3}] \cup [\frac{2}{3}, \infty)
\]

Let me know if you want further details or clarifications!

### Related Questions:
1. How do you solve absolute value equations?
2. What is the difference between solving absolute value equations and inequalities?
3. Can you graph the solution set of this inequality?
4. How do we handle inequalities with absolute values inside more complex expressions?
5. What are some real-world applications of absolute value inequalities?

**Tip:** Always check both cases when solving absolute value inequalities to capture all possible solutions.

This page provides a step-by-step solution for solving the absolute value inequality |6u| - 5 ≥ -1. Learn key algebraic concepts like absolute value properties, inequality manipulation, and how to express the solution in interval notation. Suitable for students in Grades 9-10.

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