The number of elements in the set `x in R :(absx-3)(abs(x+4))=6` is equal to

Options

A. 3
B. 4
C. 2
D. 1

Answer

To solve the equation \((|x - 3|)(|x + 4|) = 6\), we will analyze different cases based on the critical points where the expressions inside the absolute values change sign. The critical points are \(x = 3\) and \(x = -4\).

### Step-by-Step Solution:

1. **Identify Critical Points**:
The critical points are \(x = 3\) and \(x = -4\). These points divide the real number line into three intervals:
- \( (-\infty, -4) \)
- \( [-4, 3] \)
- \( [3, \infty) \)

2. **Case 1: \(x \geq 3\)**:
In this interval, both \(|x - 3|\) and \(|x + 4|\) are positive:
\[ |x - 3| = x - 3 \quad \text{and} \quad |x + 4| = x + 4 \]
Therefore, the equation becomes:
\[ (x - 3)(x + 4) = 6 \]
Expanding this gives:
\[ x^2 + x - 12 = 6 \implies x^2 + x - 18 = 0 \]
Now, we can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1 + 72}}{2} = \frac{-1 \pm \sqrt{73}}{2} \]
We need to check which of these solutions are valid in the interval \(x \geq 3\).

3. **Calculate Roots**:
\[ x = \frac{-1 + \sqrt{73}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{73}}{2} \]
The first root is approximately \(3.6\) (valid), and the second root is negative (not valid).

4. **Case 2: \(-4 \leq x < 3\)**:
Here, \(|x - 3| = 3 - x\) and \(|x + 4| = x + 4\):
\[ (3 - x)(x + 4) = 6 \]
Expanding gives:
\[ 3x + 12 - x^2 - 4x = 6 \implies -x^2 - x + 12 - 6 = 0 \implies -x^2 - x + 6 = 0 \implies x^2 + x - 6 = 0 \]
Solving this quadratic equation:
\[ x = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm 5}{2} \]
The roots are:
\[ x = 2 \quad \text{and} \quad x = -3 \]
Both values are valid in the interval \([-4, 3]\).

5. **Case 3: \(x < -4\)**:
In this case, \(|x - 3| = 3 - x\) and \(|x + 4| = -x - 4\):
\[ (3 - x)(-x - 4) = 6 \]
Expanding gives:
\[ -3x - 12 + x^2 + 4x = 6 \implies x^2 + x - 18 = 0 \]
This is the same quadratic as in Case 1, which we already solved. The valid solution here is \(x = -6\).

### Summary of Valid Solutions:
- From Case 1: \(x \approx 3.6\)
- From Case 2: \(x = 2\) and \(x = -3\)
- From Case 3: \(x = -6\)

### Final Count of Solutions:
The valid solutions are \(3.6\), \(2\), \(-3\), and \(-6\), giving us a total of **4 solutions**.

### Final Answer:
The number of elements in the set is **4**.