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.