Linear Rational Inequality

Definition

A linear rational inequality is a type of inequality that involves a rational function, where the numerator and denominator are both linear expressions. These inequalities are used to solve problems involving the comparison of two rational expressions.

5 Must Know Facts For Your Next Test

1. To solve a linear rational inequality, you must first find the critical points, which are the values of the variable where the inequality changes direction.
2. The solution set for a linear rational inequality can be expressed as a union of one or more intervals on the number line.
3. The sign of the inequality may change at the critical points, depending on the behavior of the rational function.
4. Graphing the rational function can help visualize the solution set and identify the critical points.
5. Linear rational inequalities can be used to model and solve real-world problems, such as those involving rates, ratios, and proportions.

Review Questions

• Explain the process of solving a linear rational inequality.
  • To solve a linear rational inequality, you must first find the critical points, which are the values of the variable where the inequality changes direction. This involves setting the denominator of the rational function equal to zero and solving for the variable. Once the critical points are identified, you can then determine the sign of the inequality on each interval between the critical points. The solution set is then expressed as a union of one or more intervals on the number line where the inequality is true.
• Describe the relationship between the behavior of the rational function and the solution set of a linear rational inequality.
  • The behavior of the rational function, specifically the sign changes of the function, directly affects the solution set of a linear rational inequality. The critical points of the rational function, where the denominator is equal to zero, are important because the sign of the inequality may change at these points. By analyzing the behavior of the rational function on each interval between the critical points, you can determine the solution set, which is expressed as a union of one or more intervals on the number line where the inequality is true.
• Evaluate the role of graphing in solving linear rational inequalities and explain how it can help in identifying the solution set.
  • Graphing the rational function can be a valuable tool in solving linear rational inequalities. By visualizing the graph of the rational function, you can identify the critical points, which are the $x$-intercepts of the graph. Additionally, the graph can help you determine the sign of the inequality on each interval between the critical points, as the sign of the function may change at these points. This information is crucial in determining the solution set, which is the union of one or more intervals on the number line where the inequality is true. The graph provides a clear, visual representation of the behavior of the rational function, making it easier to understand and solve the linear rational inequality.

Related terms

Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions.

Linear Expression: A linear expression is an algebraic expression that can be written in the form $ax + b$, where $a$ and $b$ are constants.

Inequality: An inequality is a mathematical statement that compares two values using symbols such as $<$, $>$, $ extbackslash leq$, or $ extbackslash geq$.