Let's solve the equation step by step.

The equation is:
\[
\left| \frac{1}{3y} - 2 \right| - 7 = 3
\]

### Step 1: Isolate the absolute value
Add 7 to both sides to remove the \(-7\) term on the left:
\[
\left| \frac{1}{3y} - 2 \right| = 3 + 7
\]
\[
\left| \frac{1}{3y} - 2 \right| = 10
\]

### Step 2: Remove the absolute value
The expression inside the absolute value can either be positive or negative. So we set up two cases:

#### Case 1: Positive expression
\[
\frac{1}{3y} - 2 = 10
\]
Solve for \(y\):
\[
\frac{1}{3y} = 10 + 2 = 12
\]
Multiply both sides by \(3y\):
\[
1 = 36y
\]
Divide by 36:
\[
y = \frac{1}{36}
\]

#### Case 2: Negative expression
\[
\frac{1}{3y} - 2 = -10
\]
Solve for \(y\):
\[
\frac{1}{3y} = -10 + 2 = -8
\]
Multiply both sides by \(3y\):
\[
1 = -24y
\]
Divide by -24:
\[
y = \frac{1}{-24} = -\frac{1}{24}
\]

### Final solutions:
The two solutions are:
\[
y = \frac{1}{36} \quad \text{or} \quad y = -\frac{1}{24}
\]

Would you like more details on any of these steps? 

Here are 5 related questions you might find interesting:
1. How do you solve absolute value inequalities?
2. What happens if there's a square root instead of an absolute value?
3. How can you check if both solutions satisfy the original equation?
4. How do you graph absolute value functions?
5. What other methods can be used to solve equations with fractions?

**Tip:** When dealing with absolute values, always remember to check for both the positive and negative cases inside the absolute value!

Learn how to solve the absolute value equation | 1/3y - 2 | - 7 = 3 with step-by-step guidance. This solution covers algebraic concepts involving absolute values, fractions, and isolating variables, making it suitable for students in Grades 8-10.

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