We can solve the equation:

$-2|x + 4| = 2|x - 1| - 10$

Step 1: Isolate the absolute value terms

Move the constant $-10$ to the other side:

$-2|x + 4| + 10 = 2|x - 1|$

This simplifies to:

$-2|x + 4| = 2|x - 1| - 10$

Divide both sides by -2:

$|x + 4| = -|x - 1| + 5$

Now we have an equation with absolute values on both sides.

Step 2: Solve by breaking down absolute values

We need to consider different cases for the absolute values.

Case 1: $x + 4 \geq 0$ and $x - 1 \geq 0$

• For this case, $|x + 4| = x + 4$ and $|x - 1| = x - 1$.

$x + 4 = -(x - 1) + 5$

Simplifying:

$x + 4 = -x + 1 + 5$

$x + 4 = -x + 6$

Add $x$ to both sides:

$2x + 4 = 6$

Subtract 4 from both sides:

$2x = 2$

Solve for $x$:

$x = 1$

This value of $x$ is valid only if $x \geq 1$. Since $x = 1$, it satisfies this condition.

Case 2: $x + 4 \geq 0$ and $x - 1 < 0$

• Here, $|x + 4| = x + 4$ and $|x - 1| = -(x - 1)$.

So the equation becomes:

$x + 4 = -(-x + 1) + 5$

Simplify:

$x + 4 = x - 1 + 5$

$x + 4 = x + 4$

This is an identity, which means it is true for all $x$, but this case is only valid when $x < 1$.

Conclusion

The solution to the equation is:

$x = 1$

Would you like more details or further explanation on any of these steps?

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Here are some follow-up questions to deepen understanding:

1. What happens when an equation has no solution involving absolute values?
2. Can you describe how to handle inequalities in absolute value equations?
3. Why does breaking down absolute values into cases work for solving these equations?
4. How would the steps differ if the equation had an additional term?
5. Can you apply this approach to solve an absolute value inequality?

Tip: Always check the conditions of each case when solving absolute value equations to ensure the solutions are valid within their respective domains.