Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
1. Equations & Inequalities
Linear Inequalities
4:39 minutes
Problem 100a
Textbook Question
Textbook QuestionIn Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. y = |2x - 5| + 1 and y > 9
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Functions
An absolute value function, such as y = |2x - 5|, measures the distance of a number from zero on the number line, resulting in non-negative outputs. This function can create two cases based on the expression inside the absolute value: one where the expression is positive and another where it is negative. Understanding how to break down these cases is essential for solving inequalities involving absolute values.
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Inequalities
Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like '>', '<', '≥', or '≤'. In this context, the inequality y > 9 indicates that we are looking for values of x that make the output of the function exceed 9. Solving inequalities often involves finding critical points and testing intervals to determine where the inequality holds true.
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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 (a, b) includes all numbers between a and b but not a and b themselves, while [a, b] includes both endpoints. This notation is crucial for succinctly expressing the solution set of inequalities.
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