The equation `x-8/(|x-3|)=3-8/(|x-3|)` has (a)only one solution (b) infinite solutions (C)no solution (d) none of these

Options

A. only one solution
B. infinite solutions
C. no solution
D. two solutions

Answer

To solve the equation \( \frac{x - 8}{|x - 3|} = 3 - \frac{8}{|x - 3|} \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{x - 8}{|x - 3|} = 3 - \frac{8}{|x - 3|} \]

### Step 2: Combine the terms
To eliminate the fraction, we can multiply both sides by \( |x - 3| \) (keeping in mind that \( |x - 3| \neq 0 \)):
\[ x - 8 = (3|x - 3|) - 8 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ x - 8 + 8 = 3|x - 3| \]
\[ x = 3|x - 3| \]

### Step 4: Analyze the absolute value
Now, we need to consider two cases based on the definition of absolute value.

**Case 1:** \( x - 3 \geq 0 \) (i.e., \( x \geq 3 \))
In this case, \( |x - 3| = x - 3 \):
\[ x = 3(x - 3) \]
\[ x = 3x - 9 \]
\[ 9 = 2x \quad \Rightarrow \quad x = \frac{9}{2} = 4.5 \]
Since \( 4.5 \geq 3 \), this solution is valid.

**Case 2:** \( x - 3 < 0 \) (i.e., \( x < 3 \))
In this case, \( |x - 3| = -(x - 3) = 3 - x \):
\[ x = 3(3 - x) \]
\[ x = 9 - 3x \]
\[ 4x = 9 \quad \Rightarrow \quad x = \frac{9}{4} = 2.25 \]
Since \( 2.25 < 3 \), this solution is also valid.

### Step 5: Check for restrictions
However, we must check if either of these solutions leads to a division by zero in the original equation. The expression \( |x - 3| \) becomes zero when \( x = 3 \).

Both solutions \( x = 4.5 \) and \( x = 2.25 \) do not equal 3, so they do not cause any issues with the denominator.

### Conclusion
Thus, the equation has two valid solutions: \( x = 4.5 \) and \( x = 2.25 \). However, since the problem asks for the type of solution, we need to consider the options given.

The correct answer is:
(b) infinite solutions.