Consider the equation: `2^(|x+1|)-2^x=|2^x-1|+1` Number of composite numbers less than 20 which are coprime with 4 satisfying the given equation is/are

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the equation \( 2^{|x+1|} - 2^x = |2^x - 1| + 1 \) and find the number of composite numbers less than 20 that are coprime with 4 and satisfy this equation, we can follow these steps:

### Step 1: Analyze the equation
The equation is given as:
\[ 2^{|x+1|} - 2^x = |2^x - 1| + 1 \]
We will consider different cases based on the value of \( x \) due to the absolute values.

### Step 2: Identify critical points
The critical points where the expressions inside the absolute values change are:
- \( x = -1 \) (for \( |x+1| \))
- \( x = 0 \) (for \( |2^x - 1| \))

### Step 3: Case 1: \( x < -1 \)
In this case, both \( |x+1| \) and \( |2^x - 1| \) are negative:
\[ |x+1| = -(x+1) \quad \text{and} \quad |2^x - 1| = -(2^x - 1) \]
Substituting these into the equation:
\[ 2^{-(x+1)} - 2^x = -(2^x - 1) + 1 \]
This simplifies to:
\[ \frac{1}{2^{x+1}} - 2^x = -2^x + 2 \]
Rearranging gives:
\[ \frac{1}{2^{x+1}} = 2 \]
This leads to:
\[ 1 = 2^{x+1} \quad \Rightarrow \quad x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
Since \( x = -1 \) is not less than -1, we discard this case.

### Step 4: Case 2: \( -1 \leq x < 0 \)
Here, \( |x+1| = x + 1 \) and \( |2^x - 1| = -(2^x - 1) \):
\[ 2^{x+1} - 2^x = -2^x + 1 + 1 \]
This simplifies to:
\[ 2^{x+1} - 2^x = -2^x + 2 \]
Rearranging gives:
\[ 2^{x+1} = 2 \]
Thus:
\[ x + 1 = 1 \quad \Rightarrow \quad x = 0 \]
Since \( x = 0 \) is not in the interval \( -1 \leq x < 0 \), we discard this case.

### Step 5: Case 3: \( x \geq 0 \)
In this case, both expressions are positive:
\[ |x+1| = x + 1 \quad \text{and} \quad |2^x - 1| = 2^x - 1 \]
Substituting gives:
\[ 2^{x+1} - 2^x = 2^x - 1 + 1 \]
This simplifies to:
\[ 2^{x+1} - 2^x = 2^x \]
Rearranging gives:
\[ 2^{x+1} = 2 \cdot 2^x \]
This holds true for all \( x \geq 0 \), meaning any \( x \geq 0 \) satisfies the equation.

### Step 6: Identify composite numbers less than 20 that are coprime with 4
The composite numbers less than 20 are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18.

Now, we check which of these are coprime with 4 (i.e., they should not share any prime factors with 4):
- 4 (not coprime)
- 6 (not coprime, shares factor 2)
- 8 (not coprime)
- 9 (coprime)
- 10 (not coprime, shares factor 2)
- 12 (not coprime)
- 14 (not coprime)
- 15 (coprime)
- 16 (not coprime)
- 18 (not coprime)

The composite numbers less than 20 that are coprime with 4 are 9 and 15.

### Final Answer
Thus, the number of composite numbers less than 20 which are coprime with 4 and satisfy the given equation is:
\[ \boxed{2} \]