12. Rational Expressions, Restrictions, and GED® Modeling

A rational expression is a fraction whose numerator or denominator contains a variable, such as:
\[ \frac{x^2-16}{x-4}. \]

Restrictions come from the denominator. Since \(x-4=0\) when \(x=4\), the expression is undefined at \(x=4\). Therefore \(x\ne4\).

You may simplify rational expressions by factoring and canceling common factors:
\[ \frac{x^2-16}{x-4}=\frac{(x-4)(x+4)}{x-4}=x+4,\qquad x\ne4. \]
The simplified expression \(x+4\) looks defined at \(x=4\), but the original expression was not. The restriction remains.

Rational expressions appear in GED®-style formulas and rates. If \(C\) dollars are shared among \(n\) people, the cost per person is:
\[ \frac{C}{n},\qquad n\ne0. \]
Zero people would make the situation impossible.

Common misconception: canceling terms instead of factors. In \(\frac{x+6}{x}\), the \(x\) in the numerator is not a separate factor, so it cannot cancel with the denominator.

Academic habit: write restrictions before simplifying. That preserves values that are not allowed.