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

Explanation

To solve the equation \( |2^x - 1| + |2^x + 1| = 2 \), we will analyze the expression based on the properties of the absolute value and the behavior of the function \( 2^x \).

### Step 1: Analyze the expression based on the value of \( 2^x \)

1. **Case 1: \( x > 0 \)**
- Here, \( 2^x > 1 \).
- Therefore, \( |2^x - 1| = 2^x - 1 \) and \( |2^x + 1| = 2^x + 1 \).
- The equation becomes:
\[ (2^x - 1) + (2^x + 1) = 2 \]
- Simplifying this gives:
\[ 2 \cdot 2^x = 2 \]
- Dividing both sides by 2:
\[ 2^x = 1 \]
- Taking the logarithm base 2:
\[ x = 0 \]

### Step 2: Check if \( x = 0 \) is a valid solution

- Since \( x = 0 \) falls under the case \( x \geq 0 \), it is indeed a valid solution.

### Step 3: Analyze the expression for \( x \leq 0 \)

2. **Case 2: \( x \leq 0 \)**
- Here, \( 2^x \leq 1 \).
- Therefore, \( |2^x - 1| = 1 - 2^x \) and \( |2^x + 1| = 2^x + 1 \).
- The equation becomes:
\[ (1 - 2^x) + (2^x + 1) = 2 \]
- Simplifying this gives:
\[ 1 - 2^x + 2^x + 1 = 2 \]
\[ 2 = 2 \]
- This is always true for any \( x \leq 0 \).

### Conclusion

- From the analysis, we find that:
- \( x = 0 \) is a specific solution.
- Any \( x \leq 0 \) is also a solution.

Thus, the complete solution set is:
\[ x \leq 0 \]