The number of real solutions of `log_(2)x+|x|=0`, is

Options

A. 0
B. 1
C. 3
D. 4

Answer

To solve the equation \( \log_{2} x + |x| = 0 \), we will analyze the two components of the equation separately and find their intersection points.

### Step 1: Rewrite the equation
We can rewrite the equation as:
\[ \log_{2} x = -|x| \]
This means we need to find where the function \( y = \log_{2} x \) intersects with \( y = -|x| \).

### Step 2: Analyze the function \( y = \log_{2} x \)
The function \( y = \log_{2} x \) is defined only for \( x > 0 \). The graph of this function passes through the point (1, 0) and increases without bound as \( x \) increases. As \( x \) approaches 0 from the right, \( y \) approaches negative infinity.

### Step 3: Analyze the function \( y = -|x| \)
The function \( y = -|x| \) can be split into two cases:
- For \( x \geq 0 \), \( y = -x \)
- For \( x < 0 \), \( y = x \)

### Step 4: Find intersections for \( x \geq 0 \)
For \( x \geq 0 \):
\[ \log_{2} x = -x \]
We need to find the intersection of \( y = \log_{2} x \) and \( y = -x \).

### Step 5: Graphical Analysis
- The graph of \( y = \log_{2} x \) is above the x-axis for \( x > 1 \) and below for \( 0 < x < 1 \).
- The graph of \( y = -x \) is a straight line with a negative slope, starting from (0, 0) and going downwards.

Since \( \log_{2} x \) is positive for \( x > 1 \) and negative for \( 0 < x < 1 \), we can conclude that there is one intersection point in the interval \( (0, 1) \).

### Step 6: Find intersections for \( x < 0 \)
For \( x < 0 \):
\[ \log_{2} x \text{ is not defined.} \]
Thus, there are no solutions for \( x < 0 \).

### Conclusion
The only real solution occurs in the interval \( (0, 1) \). Therefore, the number of real solutions to the equation \( \log_{2} x + |x| = 0 \) is:

\[ \boxed{1} \]