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.