The number of solutions of `3^(|x|)=| 2-|x||`, is
A. 0
B. 2
C. 4
D. infinite

Options

A. 0
B. 2
C. 4
D. infinite

Answer

To find the number of solutions for the equation \(3^{|x|} = |2 - |x||\), we will analyze the graphs of both sides of the equation and determine their points of intersection.

### Step 1: Understanding the Functions
We have two functions to analyze:
1. \(y_1 = 3^{|x|}\)
2. \(y_2 = |2 - |x||\)

### Step 2: Graphing \(y_1 = 3^{|x|}\)
The function \(y_1 = 3^{|x|}\) is an exponential function that is always positive. It has the following characteristics:
- At \(x = 0\), \(y_1 = 3^0 = 1\).
- As \(x\) increases or decreases, \(y_1\) increases exponentially.

### Step 3: Graphing \(y_2 = |2 - |x||\)
The function \(y_2 = |2 - |x||\) can be analyzed by breaking it down:
- For \(x \geq 0\): \(y_2 = 2 - x\) when \(x \leq 2\) and \(y_2 = x - 2\) when \(x > 2\).
- For \(x < 0\): \(y_2 = 2 + x\) when \(x \geq -2\) and \(y_2 = -2 - x\) when \(x < -2\).

### Step 4: Sketching the Graphs
1. **For \(y_1 = 3^{|x|}\)**:
- The graph rises steeply on both sides of the y-axis, starting from (0, 1).

2. **For \(y_2 = |2 - |x||**: - The graph has a V-shape with vertex at (0, 2). - It intersects the x-axis at points (-2, 0) and (2, 0). ### Step 5: Finding Points of Intersection Now we need to find the points where these two graphs intersect: - **For \(x = 0\)**: \(y_1 = 1\) and \(y_2 = 2\) (no intersection).
- **For \(0 < x < 2\)**: \(y_1\) increases while \(y_2\) decreases from 2 to 0. There will be one intersection point here.
- **For \(x = 2\)**: \(y_1 = 3^2 = 9\) and \(y_2 = 0\) (no intersection).
- **For \(x > 2\)**: \(y_1\) continues to increase while \(y_2\) increases linearly (there will be another intersection point).
- **For \(x < 0\)**: The analysis is similar. The same reasoning applies, leading to two more intersection points.

### Conclusion
Thus, we find a total of **4 intersection points** between the two graphs.

### Final Answer
The number of solutions is **4** (Option C).

---