Problem 82 - Algebra and Trigonometry

Question

What is a rational inequality?

Step-by-Step Solution

Answer

A rational inequality is an inequality with a rational expression, which is an expression of the form \(f(x) / g(x)\), where \(f(x)\) and \(g(x)\) are polynomials and \(g(x) \neq 0\). When solving rational inequalities, one must factor, identify the zeros of the numerator and denominator, divide the number line into intervals using these zeros, test each interval in the inequality, and express the solution in interval notation, excluding denominator zeros.

Step 1: Definition of Rational Inequality

A rational inequality is one that contains a rational expression. A rational expression is one that contains a variable in its denominator, and often in its numerator as well. It is of the form \(f(x) / g(x)\), where \(f(x)\) and \(g(x)\) are polynomial functions, and \(g(x) \neq 0\).

Step 2: How It Differs From Ordinary Inequalities

Unlike ordinary inequalities, rational inequalities take into account the domain of the expressions involved. The denominator of a rational expression can never be zero, so when solving rational inequalities, it's crucial to consider values of the variable that make the denominator zero and generally to exclude them from the solution set.

Step 3: Solving Rational Inequalities

To solve a rational inequality, one typically follows these steps: First, factor both the numerator and the denominator of the rational expression. Next, identify values of the variable that make the numerator and the denominator zero. Using these values, divide the number line into intervals. Test each interval in the inequality to determine whether that interval is part of the solution set. Finally, express the solution set in interval notation, excluding any values that make the denominator zero.