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.

Learn how to solve the absolute value equation -2|x+4|=3-4x with a detailed step-by-step solution. This problem covers key concepts in absolute value equations and linear equations, suitable for students in Grades 9-12.

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