We are given the equation:

$2 - 5 |4x - 10| = -43$

Step 1: Isolate the absolute value term

First, subtract 2 from both sides to simplify:

$-5 |4x - 10| = -43 - 2$

$-5 |4x - 10| = -45$

Step 2: Eliminate the negative coefficient

Now, divide both sides by -5 to eliminate the negative coefficient:

$|4x - 10| = \frac{-45}{-5}$

$|4x - 10| = 9$

Step 3: Solve for the variable inside the absolute value

The absolute value equation implies two cases:

Case 1: $4x - 10 = 9$

Solve for $x$:

$4x = 9 + 10$

$4x = 19$

$x = \frac{19}{4}$

Case 2: $4x - 10 = -9$

Solve for $x$:

$4x = -9 + 10$

$4x = 1$

$x = \frac{1}{4}$

Final Solutions:

The solutions are:

$x = \frac{19}{4} \quad \text{or} \quad x = \frac{1}{4}$

Would you like a detailed explanation of any step?

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Here are 5 related questions you might find interesting:

1. How do you solve absolute value inequalities?
2. What happens when absolute values are inside more complex expressions?
3. Can absolute value equations have no solution?
4. How can graphing help visualize solutions to absolute value equations?
5. How do you handle systems of equations involving absolute values?

Tip: Always check both cases when solving absolute value equations to avoid missing any solutions.