Problem 60 - College Algebra

Question

What is a rational inequality?

Step-by-Step Solution

Answer

A rational inequality is an inequality that compares two rational expressions, which are fractions with polynomial numerators and denominators. Solving such inequalities involves finding critical numbers and testing intervals to find the solution in interval notation.

Step 1: Understanding a rational expression

A rational expression is a fraction in which both the numerator and the denominator are polynomials. In other words, a ratio of two polynomials forms a rational expression. An example of a rational expression would be \( \frac{x^2 - 4}{x + 2} \).

Step 2: Defining Inequalities

An inequality compares two values and shows whether the first is less than, greater than, less than or equal to, or greater than or equal to the second. Examples of inequalities include expressions such as \( a > b \), \( c \leq d \), and \( x + y < z \).

Step 3: Combining the Concepts

A rational inequality then is an inequality which involves rational expressions. In other words, it compares two rational expressions to determine their relative values. An example of a rational inequality would be \( \frac{x^2 - 4}{x + 2} > 1 \). To solve such inequalities, we usually find the critical numbers (where the expressions equals to zero or undefined), make them as break points and test the intervals they create. The answers are often in interval notation.