We are given the inequality:

\[
|10 - 4y| \geq 4
\]

### Step 1: Understand the absolute value inequality

For absolute value inequalities of the form \(|A| \geq B\), where \( B \geq 0 \), we split it into two cases:
1. \( A \geq B \)
2. \( A \leq -B \)

So, applying this to our equation:

\[
10 - 4y \geq 4 \quad \text{or} \quad 10 - 4y \leq -4
\]

### Step 2: Solve each inequality separately

#### Case 1: \(10 - 4y \geq 4\)

\[
10 - 4y \geq 4
\]
Subtract 10 from both sides:
\[
-4y \geq 4 - 10
\]
\[
-4y \geq -6
\]
Now, divide by \(-4\) (remember to reverse the inequality when dividing by a negative number):
\[
y \leq \frac{-6}{-4} = \frac{3}{2}
\]
So, \(y \leq \frac{3}{2}\).

#### Case 2: \(10 - 4y \leq -4\)

\[
10 - 4y \leq -4
\]
Subtract 10 from both sides:
\[
-4y \leq -4 - 10
\]
\[
-4y \leq -14
\]
Now, divide by \(-4\) (again, reverse the inequality):
\[
y \geq \frac{-14}{-4} = \frac{7}{2}
\]
So, \(y \geq \frac{7}{2}\).

### Step 3: Combine the results

From Case 1, we have \(y \leq \frac{3}{2}\), and from Case 2, we have \(y \geq \frac{7}{2}\).

Thus, there is no overlap between these two conditions, so the solution to the inequality is:

\[
y \leq \frac{3}{2} \quad \text{or} \quad y \geq \frac{7}{2}
\]

### Final Solution:
\[
y \in (-\infty, \frac{3}{2}] \cup [\frac{7}{2}, \infty)
\]

Would you like further details on any step?

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Here are 5 related questions:
1. How do you solve absolute value equations instead of inequalities?
2. What happens when the absolute value inequality has a strict inequality, like \( |A| < B \)?
3. Can you explain how to graph absolute value inequalities on a number line?
4. What is the difference between "and" & "or" conditions in absolute value inequalities?
5. How would this solution change if the inequality was \( |10 - 4y| \leq 4 \)?

**Tip:** Always remember to flip the inequality when dividing or multiplying by a negative number!

This guide provides a step-by-step solution to the absolute value inequality |10 - 4y| ≥ 4. It covers concepts like absolute value inequalities and linear equations, suitable for students in Grades 8-10. Learn how to split the inequality into two cases and solve each one individually.

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