Here are the essential concepts you must grasp in order to answer the question correctly.
Inequalities
Inequalities express a relationship where one quantity is larger or smaller than another. In this case, the inequality 4/(x+6) > 2/(x-1) indicates that the fraction on the left must be greater than the fraction on the right. Solving inequalities often involves finding the values of the variable that satisfy this condition, which may include determining critical points and testing intervals.
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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 (a, b) includes all numbers between a and b, but not a and b themselves, while [a, b] includes a and b. This notation is essential for expressing the solution set of inequalities.
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Critical Points
Critical points are values of the variable where the expression changes its sign, often found by setting the numerator or denominator of a rational expression to zero. In the inequality 4/(x+6) > 2/(x-1), the critical points occur when x + 6 = 0 and x - 1 = 0, leading to x = -6 and x = 1. These points help to divide the number line into intervals that can be tested to determine where the inequality holds true.
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