Let's solve the equation \( \left| \frac{5X + 2}{6} \right| = \frac{7}{9} \).

### Step 1: Remove the absolute value
Since absolute value can represent both positive and negative cases, we break this equation into two cases:

1. **Case 1**: 
   \[
   \frac{5X + 2}{6} = \frac{7}{9}
   \]
2. **Case 2**: 
   \[
   \frac{5X + 2}{6} = -\frac{7}{9}
   \]

### Step 2: Solve for \(X\) in each case

#### Case 1: 
\[
\frac{5X + 2}{6} = \frac{7}{9}
\]
Cross-multiply:
\[
9(5X + 2) = 6(7)
\]
Simplifying:
\[
45X + 18 = 42
\]
Solve for \(X\):
\[
45X = 42 - 18
\]
\[
45X = 24
\]
\[
X = \frac{24}{45} = \frac{8}{15}
\]

#### Case 2:
\[
\frac{5X + 2}{6} = -\frac{7}{9}
\]
Cross-multiply:
\[
9(5X + 2) = 6(-7)
\]
Simplifying:
\[
45X + 18 = -42
\]
Solve for \(X\):
\[
45X = -42 - 18
\]
\[
45X = -60
\]
\[
X = \frac{-60}{45} = \frac{-4}{3}
\]

### Final Answer:
The solutions are:
\[
X = \frac{8}{15} \quad \text{or} \quad X = \frac{-4}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the general approach for solving absolute value equations?
2. How do you handle cases where absolute value expressions involve inequalities?
3. What are some common errors when solving equations with absolute values?
4. Can absolute value equations have no solution?
5. How do you interpret absolute value functions graphically?

**Tip:** Always check for extraneous solutions when solving absolute value equations, especially in more complex contexts.

This detailed solution walks you through solving the absolute value equation |5X + 2/6| = 7/9, with steps covering absolute value properties, cross-multiplication, and linear equation solving. Learn to handle equations involving absolute values with two cases, suitable for Grades 8-10.

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