Problem 59 - College Algebra Essentials

Question

Simplify each complex rational expression. $$\frac{\frac{x}{3}-1}{x-3}$$

Step-by-Step Solution

Answer

The answer is \( \frac{1}{3}\).

Step 1: Identify the Least Common Multiple

First, identify the least common multiple (LCM) of the denominators in the complex rational expression. In this case, the denominators are \(3\) and \(1\), so the LCM is \(3\).

Step 2: Multiply by LCM

Multiply (or distribute) the LCM to every fraction in the numerator and denominator. This gives \(\frac{3*x/3 - 3*1}{3*x -3*3}\).

Step 3: Simplify the Expression

Now simplify each term. Remember that in the case of \((3*x/3)\), the \(3\) in the numerator and in the denominator cancels out to equal 1. This results in \(x\) only. So the new equation becomes \( \frac{x - 3}{3*x - 9}\).

Step 4: Factor the Denominator

Factoring the denominator would give us 3 times the quantity of \(x-3\), such that \(\frac{x - 3}{3*(x-3)}\).

Step 5: Further Simplify the Expression

Now, cancel out the \(x-3\) in the numerator with the \(x-3\) in the denominator. This will leave \(\frac{1}{3}\) as the simplified complex fraction.