Numberof integral solution of `2x -1 = |x +7|` is

Answer

To solve the equation \(2x - 1 = |x + 7|\) for integral solutions, we need to consider the definition of the absolute value and break it down into two cases based on the value of \(x\).

### Step 1: Identify the cases for the absolute value
The absolute value function \( |x + 7| \) can be expressed as:
- \( x + 7 \) when \( x + 7 \geq 0 \) (i.e., \( x \geq -7 \))
- \( -(x + 7) \) when \( x + 7 < 0 \) (i.e., \( x < -7 \))

### Step 2: Case 1 - \( x < -7 \)
For this case, we rewrite the equation:
\[ 2x - 1 = -(x + 7) \]
This simplifies to:
\[ 2x - 1 = -x - 7 \]
Now, rearranging gives:
\[ 2x + x = -7 + 1 \]
\[ 3x = -6 \]
Dividing both sides by 3:
\[ x = -2 \]
However, \( x = -2 \) does not satisfy the condition \( x < -7 \). Therefore, there are no valid solutions from this case.

### Step 3: Case 2 - \( x \geq -7 \)
For this case, we rewrite the equation:
\[ 2x - 1 = x + 7 \]
This simplifies to:
\[ 2x - 1 = x + 7 \]
Rearranging gives:
\[ 2x - x = 7 + 1 \]
\[ x = 8 \]
Now, we check if \( x = 8 \) satisfies the condition \( x \geq -7 \). Since \( 8 \geq -7 \) is true, this is a valid solution.

### Step 4: Conclusion
The only integral solution we found is \( x = 8 \). Therefore, the number of integral solutions to the equation \( 2x - 1 = |x + 7| \) is:

\[ \boxed{1} \]