Let S denote the set of all values of the parameter a for which `x+sqrt(x^(2))=a` has no solution, them S equals

Options

A. (-1,1)
B. `(-oo,-1)`
C. `[-1,oo)`
D. `(0,oo)`

Explanation

To solve the problem, we need to analyze the equation \( x + \sqrt{x^2} = a \) and determine the values of \( a \) for which there are no solutions.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
The equation \( x + \sqrt{x^2} = a \) can be simplified. Since \( \sqrt{x^2} = |x| \), we can rewrite the equation as:
\[ x + |x| = a \]

2. **Consider Cases for \( |x| \)**:
The absolute value function \( |x| \) behaves differently based on the sign of \( x \). We will consider two cases:
- **Case 1**: \( x \geq 0 \) (non-negative values)
- **Case 2**: \( x < 0 \) (negative values)

3. **Case 1: \( x \geq 0 \)**:
In this case, \( |x| = x \). Thus, the equation becomes:
\[ x + x = a \implies 2x = a \implies x = \frac{a}{2} \]
Here, \( x \) must be non-negative, which gives us the condition:
\[ \frac{a}{2} \geq 0 \implies a \geq 0 \]

4. **Case 2: \( x < 0 \)**:
In this case, \( |x| = -x \). Thus, the equation becomes:
\[ x - x = a \implies 0 = a \]
This means that when \( x < 0 \), the only solution is when \( a = 0 \).

5. **Summary of Cases**:
- For \( a \geq 0 \): There are solutions \( x = \frac{a}{2} \) when \( a > 0 \).
- For \( a = 0 \): The solution is \( x = 0 \).
- For \( a < 0 \): There are no solutions since \( x + |x| \) cannot yield a negative value.

6. **Conclusion**:
The set \( S \) of all values of \( a \) for which the equation has no solution is:
\[ S = (-\infty, 0) \]

### Final Answer:
The set \( S \) equals \( (-\infty, 0) \).