Consider the equation: `2^(|x+1|)-2^x=|2^x-1|+1` Number of integers less than 15 satisfying the equation are

Options

A. 14
B. 15
C. 16
D. 17

Answer

To solve the equation \(2^{|x+1|} - 2^x = |2^x - 1| + 1\), we will analyze it by considering different cases based on the value of \(x\).

### Step 1: Identify critical points
The critical points occur when the expressions inside the absolute values change. Here, we have:
1. \(x + 1 = 0 \Rightarrow x = -1\)
2. \(2^x - 1 = 0 \Rightarrow 2^x = 1 \Rightarrow x = 0\)

Thus, we will consider three cases based on these critical points:
- Case 1: \(x < -1\)
- Case 2: \(-1 \leq x < 0\)
- Case 3: \(x \geq 0\)

### Step 2: Case 1: \(x < -1\)
In this case:
- \(|x + 1| = -(x + 1) = -x - 1\)
- \(|2^x - 1| = 1 - 2^x\)

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

### Step 3: Case 2: \(-1 \leq x < 0\)
In this case:
- \(|x + 1| = x + 1\)
- \(|2^x - 1| = 1 - 2^x\)

Substituting these into the equation gives:
\[ 2^{x+1} - 2^x = (1 - 2^x) + 1 \]
This simplifies to:
\[ 2^{x+1} - 2^x = 2 - 2^x \]
Rearranging terms:
\[ 2^{x+1} = 2 \]
This leads to:
\[ x + 1 = 1 \Rightarrow x = 0 \]
However, \(x = 0\) is not included in this case, so there are no solutions from this case.

### Step 4: Case 3: \(x \geq 0\)
In this case:
- \(|x + 1| = x + 1\)
- \(|2^x - 1| = 2^x - 1\)

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

### Step 5: Count the integer solutions
Now we need to count the integer solutions less than 15:
- From Case 1, we have \(x = -2\).
- From Case 3, all integers \(x = 0, 1, 2, \ldots, 14\) are solutions.

Counting these:
- From Case 1: 1 solution (\(-2\))
- From Case 3: 15 solutions (0 through 14)

### Total number of integer solutions
Total = \(1 + 15 = 16\).

Thus, the number of integers less than 15 satisfying the equation is **16**.