Consider the equation: `2^(|x+1|)-2^x=|2^x-1|+1` The least value of `x` satisfying is a. 0 b. 2 c. 4 d. None of these

Options

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

Answer

To solve the equation \(2^{|x+1|} - 2^x = |2^x - 1| + 1\), we will analyze different cases based on the values of \(x\) that affect the absolute values.

### Step 1: Identify critical points for absolute values
The absolute values in the equation are affected by the expressions \(x + 1\) and \(2^x - 1\). We find the critical points:
- \(x + 1 = 0 \Rightarrow x = -1\)
- \(2^x - 1 = 0 \Rightarrow 2^x = 1 \Rightarrow x = 0\)

These points divide the number line into intervals:
1. \(x < -1\)
2. \(-1 \leq x < 0\)
3. \(x \geq 0\)

### Step 2: Case 1: \(x < -1\)
In this case, both expressions inside the absolute values are negative:
- \(|x + 1| = -(x + 1) = -x - 1\)
- \(|2^x - 1| = -(2^x - 1) = 1 - 2^x\)

Substituting these into the equation:
\[ 2^{-x-1} - 2^x = (1 - 2^x) + 1 \]
This simplifies to:
\[ 2^{-x-1} - 2^x = 2 - 2^x \]
Rearranging gives:
\[ 2^{-x-1} = 2 \]
This implies:
\[ -x - 1 = 1 \Rightarrow -x = 2 \Rightarrow x = -2 \]

### Step 3: Verify the solution in this case
We check if \(x = -2\) satisfies the original equation:
- LHS: \(2^{|-2 + 1|} - 2^{-2} = 2^{1} - \frac{1}{4} = 2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}\)
- RHS: \(|2^{-2} - 1| + 1 = |\frac{1}{4} - 1| + 1 = |\frac{-3}{4}| + 1 = \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}\)

Both sides are equal, confirming \(x = -2\) is a valid solution.

### Step 4: Case 2: \(-1 \leq x < 0\)
Here, we have:
- \(|x + 1| = x + 1\)
- \(|2^x - 1| = 1 - 2^x\)

Substituting into the equation:
\[ 2^{x+1} - 2^x = (1 - 2^x) + 1 \]
This simplifies to:
\[ 2^{x+1} - 2^x = 2 - 2^x \]
Rearranging gives:
\[ 2^{x+1} = 2 \]
This implies:
\[ x + 1 = 1 \Rightarrow x = 0 \]

### Step 5: Verify the solution in this case
Check if \(x = 0\) satisfies the original equation:
- LHS: \(2^{|0 + 1|} - 2^0 = 2^{1} - 1 = 2 - 1 = 1\)
- RHS: \(|2^0 - 1| + 1 = |1 - 1| + 1 = 0 + 1 = 1\)

Both sides are equal, confirming \(x = 0\) is a valid solution.

### Step 6: Case 3: \(x \geq 0\)
In this case, both expressions inside the absolute values are positive:
- \(|x + 1| = x + 1\)
- \(|2^x - 1| = 2^x - 1\)

Substituting into the equation:
\[ 2^{x+1} - 2^x = (2^x - 1) + 1 \]
This simplifies to:
\[ 2^{x+1} - 2^x = 2^x \]
Rearranging gives:
\[ 2^{x+1} = 2 \cdot 2^x \]
This is always true for \(x \geq 0\).

### Conclusion
The least value of \(x\) satisfying the equation is \(-2\) from Case 1, but since this is not among the options provided, the answer is:
**d. None of these**