Problem 101 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.

Step-by-Step Solution

Answer

The given statement does not make sense because the resulting expression after subtracting the presented fractions is not a constant, but a function dependent on 'x'. The result is \(\frac{-2x +2}{x -1}\), which clearly depends on 'x' and is, hence, not a constant.

Step 1: Understand the given statement

The given statement says that upon subtracting \(\frac{3x - 5}{x - 1}\) from \(\frac{x - 3}{x - 1}\), we get a constant. A constant is a value that doesn't change irrespective of the value of the variable involved, in this case, x.

Step 2: Subtract the two fractions

Both fractions have the same denominator, which is \(x - 1\). When we subtract two fractions with the same denominator, we subtract the numerators while the denominator remains the same, thus \[\frac{(x-3) - (3x - 5)}{x - 1} = \frac{x - 3 - 3x +5}{x - 1} = \frac{-2x + 2}{x - 1}.\]

Step 3: Analyze the result

Inspecting the outcome \(\frac{-2x + 2}{x - 1}\), it clearly depends on the value of 'x' and is not a constant because the numerator contains an 'x' term. Therefore, the given statement does not make sense.