Problem 43 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x^{2}-4}{x} \div \frac{x+2}{x-2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{x-2}{x}\), with \(x \neq 2\) and \(x \neq 0\)

Step 1: Simplify the Expressions

The numerator in the first expression \(x^{2}-4\) can be factored into \((x+2)(x-2)\) using difference of squares formula. So, the first expression becomes \(\frac{(x+2)(x-2)}{x}\). The second expression remains \(\frac{x+2}{x-2}\).

Step 2: Convert the Division to Multiplication

Division of rational expressions is similar to division of fractions. To divide, multiply by the reciprocal of the divisor. Therefore, \[\frac{(x+2)(x-2)}{x} \div \frac{x+2}{x-2}\] can be rewritten as \[\frac{(x+2)(x-2)}{x} \cdot \frac{x-2}{x+2}\]

Step 3: Simplify the Expression

(x+2) and (x-2) in the numerator and denominator gets cancelled out. The simplified expression is \[\frac{x-2}{x}\]

Step 4: State the Restrictions

The denominator of a fraction cannot be zero, therefore the possible restrictions are \(x \neq 2\), coming from \((x-2)\) in the denominator, and \(x \neq 0\), from x in the denominator of the initial expression.