We can solve the equation:

\[
|3x - 10| - 2 = 5x
\]

### Step 1: Isolate the absolute value term.
Add 2 to both sides:

\[
|3x - 10| = 5x + 2
\]

### Step 2: Solve the absolute value equation.
For an absolute value equation, we have two cases to consider:

1. **Positive Case:**
   \[
   3x - 10 = 5x + 2
   \]
   Solve for \(x\):

   \[
   3x - 5x = 2 + 10
   \]
   \[
   -2x = 12
   \]
   \[
   x = -6
   \]

2. **Negative Case:**
   \[
   3x - 10 = -(5x + 2)
   \]
   Simplify the right-hand side:

   \[
   3x - 10 = -5x - 2
   \]
   Solve for \(x\):

   \[
   3x + 5x = -2 + 10
   \]
   \[
   8x = 8
   \]
   \[
   x = 1
   \]

### Step 3: Check both solutions.
For \(x = -6\):
\[
|3(-6) - 10| - 2 = 5(-6)
\]
\[
|-18 - 10| - 2 = -30
\]
\[
| -28 | - 2 = -30
\]
\[
28 - 2 = -30 \quad \text{(False)}
\]
Thus, \(x = -6\) is **not a valid solution**.

For \(x = 1\):
\[
|3(1) - 10| - 2 = 5(1)
\]
\[
|3 - 10| - 2 = 5
\]
\[
| -7 | - 2 = 5
\]
\[
7 - 2 = 5 \quad \text{(True)}
\]
Thus, \(x = 1\) is the **valid solution**.

### Final Answer:
\[
x = 1
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand this concept:
1. How do you solve an equation with two absolute value terms?
2. What happens if the solution for \(x\) involves no valid solutions?
3. How do you graph the function \(f(x) = |3x - 10| - 2\)?
4. How do you solve quadratic equations involving absolute values?
5. How do transformations of absolute value functions affect their graph?

**Tip:** Always check your solutions when solving absolute value equations, as not all will satisfy the original equation.

Learn how to solve the absolute value equation |3x−10|−2=5x with a detailed, step-by-step solution. This problem involves isolating the absolute value term and considering both the positive and negative cases, suitable for students in Grades 8-10.

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