Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Inequalities
Rational inequalities involve expressions that contain a rational function, typically in the form of a fraction where the numerator and denominator are polynomials. To solve these inequalities, one must determine where the rational expression is greater than, less than, or equal to a certain value. This often requires finding critical points where the expression is undefined or equals zero, and testing intervals to establish the solution set.
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Rationalizing Denominators
Interval Notation
Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2. Understanding how to express solutions in this format is essential for clearly communicating the results of inequalities.
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Critical Points
Critical points are values of the variable where the rational expression is either zero or undefined. These points are crucial for solving rational inequalities as they divide the number line into intervals that can be tested for the inequality's truth. In the given inequality, finding where the expression equals 1 or is undefined will help identify these critical points, allowing for a systematic approach to determine the solution set.
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