Number of solution of `(absx-3)abs(x+4)=6`

Answer

To solve the equation \((|x| - 3)|x + 4| = 6\), we will consider different cases based on the values of \(x\) because of the absolute values involved.

### Step 1: Identify Cases Based on Absolute Values

We need to consider the following cases based on the expressions inside the absolute values:

1. **Case 1:** \(x < -4\)
2. **Case 2:** \(-4 \leq x < 3\)
3. **Case 3:** \(x \geq 3\)

### Step 2: Solve Case 1 (\(x < -4\))

In this case, both \(|x| = -x\) and \(|x + 4| = -(x + 4)\). Therefore, the equation becomes:

\[ (-x - 3)(-x - 4) = 6 \]

Simplifying this:

\[ (x + 3)(x + 4) = 6 \]

Expanding:

\[ x^2 + 7x + 12 = 6 \]

Rearranging gives:

\[ x^2 + 7x + 6 = 0 \]

Now, we can factor this quadratic:

\[ (x + 1)(x + 6) = 0 \]

Thus, the solutions are:

\[ x = -1 \quad \text{and} \quad x = -6 \]

Since we are in Case 1 (\(x < -4\)), only \(x = -6\) is valid.

### Step 3: Solve Case 2 (\(-4 \leq x < 3\))

In this case, \(|x| = -x\) and \(|x + 4| = x + 4\). Therefore, the equation becomes:

\[ (-x - 3)(x + 4) = 6 \]

Expanding this:

\[ -x^2 - 4x - 3x - 12 = 6 \]

Simplifying gives:

\[ -x^2 - 7x - 12 = 6 \]

Rearranging gives:

\[ -x^2 - 7x - 18 = 0 \quad \Rightarrow \quad x^2 + 7x + 18 = 0 \]

Now, we calculate the discriminant:

\[ D = 7^2 - 4 \cdot 1 \cdot 18 = 49 - 72 = -23 \]

Since the discriminant is negative, there are no real solutions in this case.

### Step 4: Solve Case 3 (\(x \geq 3\))

In this case, both \(|x| = x\) and \(|x + 4| = x + 4\). Therefore, the equation becomes:

\[ (x - 3)(x + 4) = 6 \]

Expanding this:

\[ x^2 + 4x - 3x - 12 = 6 \]

Simplifying gives:

\[ x^2 + x - 12 = 6 \]

Rearranging gives:

\[ x^2 + x - 18 = 0 \]

Now, we can factor this quadratic:

\[ (x - 3)(x + 6) = 0 \]

Thus, the solutions are:

\[ x = 3 \quad \text{and} \quad x = -6 \]

Since we are in Case 3 (\(x \geq 3\)), only \(x = 3\) is valid.

### Step 5: Summary of Solutions

From all cases, we found the valid solutions:

1. From Case 1: \(x = -6\)
2. From Case 3: \(x = 3\)

Thus, the total number of solutions is **2**.