Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Inequalities
Rational inequalities involve expressions that are ratios of polynomials set in an inequality format. To solve them, one must determine where the rational expression is greater than or less than a certain value. This often requires finding critical points where the expression is undefined or equal to zero, and then testing intervals to see where the inequality holds true.
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Rationalizing Denominators
Interval Notation
Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (1, 5] includes all numbers greater than 1 and up to 5, including 5 but not 1. This notation is essential for clearly expressing the solution set 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 in solving rational inequalities as they divide the number line into intervals that can be tested for the inequality's validity. Identifying these points allows for a systematic approach to determine where the inequality holds true across the different intervals.
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