Question

Carmen was finding the solutions to the equation 1/x-3 x/x-2=1/(x-3)(x-2)

Answer 1

The solution to the equation
`\( (1)/(x-3) \cdot (x)/(x-2) = (1)/((x-3)(x-2)) \) is \( x = 1 \)`.

To solve the equation
`\( (1)/(x-3) \cdot (x)/(x-2) = (1)/((x-3)(x-2)) \)`, we shall first clarify the equation to ensure we understand the terms correctly:

The equation is
`\( \left((1)/(x-3)\right) \cdot \left((x)/(x-2)\right) = (1)/((x-3)(x-2)) \)`.

To solve this equation, we will follow a step-by-step process:

1. **Multiply both sides of the equation by
`\((x-3)(x-2)\)`** to eliminate the denominators:

`\( (x-3)(x-2) \cdot \left((1)/(x-3) \cdot (x)/(x-2)\right) = (x-3)(x-2) \cdot (1)/((x-3)(x-2)) \)`

2. Simplify both sides:
On the left side,
`\((x-3)\)` will cancel out one
`\((x-3)\)` in the numerator, and
`\((x-2)\)` will cancel out one
`\((x-2)\)` in the numerator, leaving us with:

`\( x = 1 \)`

3. We see that once we simplify, what remains is simply
`\( x = 1 \)`. However, we must be careful to check whether this solution is valid in the context of the original equation.

4. **Check for extraneous solutions.**
The original equation has denominators that include
`\((x-3)\)` and
`\((x-2)\)`. For these denominators to be valid (i.e., we don't divide by zero),
`\( x \)` cannot be 3 or 2.

Since
`\( x = 1 \)` is not equal to 2 or 3, it does not make any denominator zero. Thus, the solution
`\( x = 1 \)` is valid.

Therefore, the solution to the equation
`\( (1)/(x-3) \cdot (x)/(x-2) = (1)/((x-3)(x-2)) \)` is
`\( x = 1 \)`.