Rational Inequality

Definition

A rational inequality is an inequality that involves a rational expression, which is a fraction with a polynomial in the numerator and a non-zero polynomial in the denominator. Solving rational inequalities requires analyzing the sign of the rational expression over the given interval.

5 Must Know Facts For Your Next Test

1. Rational inequalities can be solved by first finding the critical points, which are the values of the variable where the rational expression is undefined or changes sign.
2. The sign of the rational expression must be analyzed on each interval between the critical points to determine the solution set.
3. Graphing the rational expression can be a helpful tool in visualizing the solution set for a rational inequality.
4. Simplifying the rational expression by factoring the numerator and denominator can make solving the inequality easier.
5. Rational inequalities can have one, two, or infinitely many solutions, depending on the structure of the rational expression.

Review Questions

• Explain the steps involved in solving a rational inequality.
  • To solve a rational inequality, the first step is to identify the critical points, which are the values of the variable where the rational expression is undefined or changes sign. Next, the sign of the rational expression must be analyzed on each interval between the critical points to determine the solution set. This can be done by testing a sample point in each interval and observing the sign of the expression. Finally, the solution set can be expressed using interval notation or a graph of the rational expression.
• Describe how the structure of a rational expression affects the solution set of a rational inequality.
  • The structure of the rational expression, including the degree and factorization of the numerator and denominator polynomials, can significantly impact the solution set of a rational inequality. For example, a rational expression with a linear numerator and denominator will have at most one solution, while a rational expression with a quadratic numerator and denominator may have two or more solutions. Additionally, the presence of factors in the numerator or denominator that change sign can lead to multiple solution intervals or even an infinite number of solutions.
• Analyze how the process of simplifying a rational expression can aid in solving a rational inequality.
  • Simplifying a rational expression by factoring the numerator and denominator can make the process of solving a rational inequality much more straightforward. By factoring, the critical points of the expression become more evident, as they correspond to the zeros of the numerator and denominator. Additionally, factoring can reveal the sign changes of the rational expression, which are crucial in determining the solution set. Furthermore, a simplified rational expression is often easier to graph, providing a visual aid in understanding the behavior of the inequality and its solution set.

Related terms

Rational Expression: A rational expression is a fraction where both the numerator and denominator are polynomials.

Polynomial: A polynomial is an expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.

Sign Analysis: Sign analysis is the process of determining the sign (positive, negative, or zero) of a rational expression over a given interval.