Let's solve the equation:

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

### Step 1: Isolate the absolute value
First, add 2 to both sides of the equation:
\[
|3x - 10| = 5x + 2
\]

### Step 2: Consider the two cases of the absolute value

#### Case 1: \(3x - 10 \geq 0\)
If \(3x - 10 \geq 0\), then \(|3x - 10| = 3x - 10\). The equation becomes:
\[
3x - 10 = 5x + 2
\]
Now solve for \(x\):
\[
3x - 5x = 2 + 10
\]
\[
-2x = 12
\]
\[
x = -6
\]

Next, check if \(x = -6\) satisfies the condition \(3x - 10 \geq 0\):
\[
3(-6) - 10 = -18 - 10 = -28
\]
Since \(-28 < 0\), this value doesn't satisfy the condition for Case 1. Therefore, \(x = -6\) is not a valid solution here.

#### Case 2: \(3x - 10 < 0\)
If \(3x - 10 < 0\), then \(|3x - 10| = -(3x - 10) = -3x + 10\). The equation now becomes:
\[
-3x + 10 = 5x + 2
\]
Solve for \(x\):
\[
10 - 2 = 5x + 3x
\]
\[
8 = 8x
\]
\[
x = 1
\]

Check if \(x = 1\) satisfies \(3x - 10 < 0\):
\[
3(1) - 10 = 3 - 10 = -7
\]
Since \(-7 < 0\), \(x = 1\) is valid.

### Conclusion
The only valid solution is:
\[
\boxed{x = 1}
\]

Would you like more details or have any questions?

### 5 Related Questions
1. How would the solution change if the absolute value equation was \(|3x - 10| + 2 = 5x\)?
2. Can you solve \( |x + 4| = 3x - 2 \)?
3. What are the general steps for solving absolute value equations?
4. How would we solve the inequality \( |3x - 10| \leq 5x + 2 \)?
5. Can you graph \(y = |3x - 10|\) and \(y = 5x + 2\) to visualize their intersection?

### Tip
Always check the validity of your solutions with absolute value equations, as not all solutions obtained will necessarily satisfy the original conditions.

Learn how to solve the absolute value equation ∣3x−10∣−2=5x with a step-by-step approach. This problem involves isolating the absolute value, handling two cases, and checking the validity of solutions. Suitable for students in Grades 9-11 studying algebra and absolute value equations.

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