Find the number of solutions of ` |x|*3^(|x|) = 1`.

Explanation

To find the number of solutions for the equation \( |x| \cdot 3^{|x|} = 1 \), we can break it down into two cases based on the definition of the absolute value.

### Step 1: Define the cases based on the absolute value
1. **Case 1**: \( x \geq 0 \)
- Here, \( |x| = x \). The equation becomes:
\[ x \cdot 3^x = 1 \]

2. **Case 2**: \( x < 0 \)
- Here, \( |x| = -x \). The equation becomes:
\[ -x \cdot 3^{-x} = 1 \quad \text{or} \quad x \cdot 3^{-x} = -1 \]

### Step 2: Analyze Case 1
For \( x \geq 0 \):
- We need to solve \( x \cdot 3^x = 1 \).
- Define the function \( f(x) = x \cdot 3^x \).
- We can analyze the function \( f(x) \):
- As \( x \) approaches 0, \( f(0) = 0 \).
- As \( x \) increases, \( f(x) \) increases because both \( x \) and \( 3^x \) are increasing functions.

To find the intersection with \( y = 1 \):
- We can check values:
- \( f(0) = 0 \)
- \( f(1) = 1 \cdot 3^1 = 3 \) (which is greater than 1)

Since \( f(x) \) is continuous and strictly increasing, there is exactly **one solution** in the interval \( [0, 1] \).

### Step 3: Analyze Case 2
For \( x < 0 \):
- We need to solve \( x \cdot 3^{-x} = -1 \).
- Define the function \( g(x) = x \cdot 3^{-x} \).
- Analyze the function \( g(x) \):
- As \( x \) approaches 0 from the left, \( g(x) \) approaches 0.
- As \( x \) decreases (becomes more negative), \( g(x) \) also decreases because \( 3^{-x} \) increases rapidly.

To find the intersection with \( y = -1 \):
- Check values:
- As \( x \) approaches -1, \( g(-1) = -1 \cdot 3^{1} = -3 \).
- As \( x \) approaches 0 from the left, \( g(x) \) approaches 0.

Since \( g(x) \) is continuous and strictly decreasing, there is exactly **one solution** in the interval \( (-\infty, 0) \).

### Step 4: Conclusion
- From Case 1, we found **1 solution** for \( x \geq 0 \).
- From Case 2, we found **1 solution** for \( x < 0 \).

Thus, the total number of solutions to the equation \( |x| \cdot 3^{|x|} = 1 \) is:
\[ \text{Total Solutions} = 1 + 1 = 2 \]