Question

The number of solution(s) of the equation \[ x |x + 4| + 3 |x + 2| = 0 \] is/are equal to:

Hint:

When solving absolute value equations, break the problem into cases based on the sign of the expression inside the absolute value.

Answer 1

Correct Answer: 3

Step 1: Understanding the absolute values.
We need to consider different cases based on the values inside the absolute value functions. **Case 1: \( x \geq -2 \).**
For \( x \geq -2 \), the absolute value functions become: \[ |x + 4| = x + 4, \quad |x + 2| = x + 2 \] Substituting into the equation: \[ x(x + 4) + 3(x + 2) = 0 \] Simplify and solve the resulting quadratic equation. **Case 2: \( -4 \leq x<-2 \).**
For \( -4 \leq x<-2 \), we have: \[ |x + 4| = x + 4, \quad |x + 2| = -(x + 2) \] Substitute and solve the equation. **Case 3: \( x<-4 \).**
For \( x<-4 \), we have: \[ |x + 4| = -(x + 4), \quad |x + 2| = -(x + 2) \] Substitute and solve the equation. Step 2: Conclusion.
After solving all the cases, we find three solutions for \( x \). Final Answer: \[ \boxed{3} \]