The equation to solve is:

$2 - 5 \left| 4x - 10 \right| = -43$

Step 1: Isolate the absolute value term.

First, subtract 2 from both sides of the equation:

$-5 \left| 4x - 10 \right| = -43 - 2$

$-5 \left| 4x - 10 \right| = -45$

Step 2: Divide by -5 to get rid of the coefficient.

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

$\left| 4x - 10 \right| = 9$

Step 3: Solve the absolute value equation.

Recall that if $\left| A \right| = B$, then $A = B$ or $A = -B$.

So we have 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 Answer:

The two solutions are:

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

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Do you need more details or have any questions?

Related Questions:

1. What is the general process for solving absolute value equations?
2. How do you verify solutions to absolute value equations?
3. Can absolute value equations ever have no solution?
4. How does changing the coefficient of the absolute value affect the solution?
5. What happens if the right-hand side of an absolute value equation is negative?

Tip: Always check if a solution satisfies the original equation by substituting back into it.