To solve the equation $-2|x+4| = 3 - 4x$, we need to consider the two cases for the absolute value $|x + 4|$.

Case 1: $x + 4 \geq 0$ (i.e., $x \geq -4$)

In this case, $|x + 4| = x + 4$. The equation becomes: $-2(x + 4) = 3 - 4x$ Simplifying: $-2x - 8 = 3 - 4x$ Add $4x$ to both sides: $2x - 8 = 3$ Add 8 to both sides: $2x = 11$ Divide by 2: $x = \frac{11}{2} = 5.5$

Since $x = 5.5 \geq -4$, this solution is valid.

Case 2: $x + 4 < 0$ (i.e., $x < -4$)

In this case, $|x + 4| = -(x + 4)$. The equation becomes: $-2(-x - 4) = 3 - 4x$ Simplifying: $2x + 8 = 3 - 4x$ Add $4x$ to both sides: $6x + 8 = 3$ Subtract 8 from both sides: $6x = -5$ Divide by 6: $x = -\frac{5}{6}$

Since $-\frac{5}{6}$ is not less than $-4$, this solution is not valid.

Final Answer:

The only valid solution is $x = 5.5$.

Would you like more details or have any questions?

Here are 5 related questions to further explore the topic:

1. What happens if the absolute value function is shifted differently?
2. How would the solution change if the coefficients were different?
3. Can we use a graphical method to solve the absolute value equation?
4. How do we solve absolute value inequalities?
5. What are the properties of absolute value functions in equations?

Tip: Always check the conditions for validity when solving absolute value equations, as they often create multiple cases to consider.