Problem 66 - College Algebra

Question

Simplify each complex rational expression. $$\frac{x-3}{x-\frac{3}{x-2}}$$

Step-by-Step Solution

Answer

\(\frac{1}{x+1}\)

Step 1: Identify the Complex Fraction

The given expression is a complex fraction, which is a fraction where the numerator or the denominator or both contain a fraction.

Step 2: Simplify the denominator

To simplify the expression, it's easier to deal with the denominator first. The denominator is \(x-\frac{3}{x-2}\). The aim is to get rid of the fraction in the denominator. This can be done by multiplying everything by \(x-2\) which is the denominator of the sub-fraction. That will clear out the sub-fraction. Doing this yields: \(x(x-2) - 3\). After expanding and simplifying, you get: \(x^2-2x-3\). Now the expression is looking like this: \(\frac{x-3}{x^2-2x-3}\)

Step 3: Factorize the denominator

Factorizing the quadratic equation in the denominator gives \((x-3)(x+1)\). The whole expression now looks like \(\frac{x-3}{(x-3)(x+1)}\)

Step 4: Simplify the Fraction

Now, you can simplify the whole fraction by cancelling out the common factor in both numerator and denominator, which in this case is \((x-3)\). After cancelling out the common factor, you are left with: \(\frac{1}{x+1}\).