Solve `|2^x-1|+|2^x+1|=2`

Explanation

To solve the equation \( |2^x - 1| + |2^x + 1| = 2 \), we will analyze the absolute values and break the problem into cases based on the value of \( 2^x \).

### Step 1: Analyze the absolute values
The expression \( |2^x - 1| + |2^x + 1| \) can be broken down into different cases based on the value of \( 2^x \).

1. **Case 1**: \( 2^x - 1 \geq 0 \) (i.e., \( 2^x \geq 1 \) or \( x \geq 0 \))
- Here, \( |2^x - 1| = 2^x - 1 \)
- And \( |2^x + 1| = 2^x + 1 \)
- Thus, the equation becomes:
\[ (2^x - 1) + (2^x + 1) = 2 \]
Simplifying this gives:
\[ 2 \cdot 2^x = 2 \implies 2^x = 1 \implies x = 0 \]

2. **Case 2**: \( 2^x - 1 < 0 \) (i.e., \( 2^x < 1 \) or \( x < 0 \))
- Here, \( |2^x - 1| = -(2^x - 1) = -2^x + 1 \)
- And \( |2^x + 1| = 2^x + 1 \)
- Thus, the equation becomes:
\[ (-2^x + 1) + (2^x + 1) = 2 \]
Simplifying this gives:
\[ 2 = 2 \]
This is always true for \( x < 0 \).

### Step 2: Combine the results
From Case 1, we found that \( x = 0 \) is a solution. From Case 2, we found that any \( x < 0 \) is also a solution.

### Final Solution
The complete solution set for the equation \( |2^x - 1| + |2^x + 1| = 2 \) is:
\[ x \leq 0 \]