Domanda del libro di testo
Solve each equation in Exercises 83–108 by the method of your choice.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 2, Problema 100
Use interval notation to represent all values of x satisfying the given conditions. y = |2x - 5| + 1 and y > 9
Guida verificata passo dopo passo1
Start by isolating the absolute value expression in the inequality: given \( y = |2x - 5| + 1 \) and \( y > 9 \), substitute \( y \) to get \( |2x - 5| + 1 > 9 \).
Subtract 1 from both sides to simplify the inequality: \( |2x - 5| > 8 \).
Recall that for an absolute value inequality \( |A| > B \) where \( B > 0 \), the solution splits into two cases: \( A > B \) or \( A < -B \). Apply this to get two inequalities: \( 2x - 5 > 8 \) or \( 2x - 5 < -8 \).
Solve each inequality separately:
- For \( 2x - 5 > 8 \), add 5 to both sides to get \( 2x > 13 \), then divide by 2 to find \( x > \frac{13}{2} \).
- For \( 2x - 5 < -8 \), add 5 to both sides to get \( 2x < -3 \), then divide by 2 to find \( x < -\frac{3}{2} \).
Express the solution in interval notation by combining both parts: \( (-\infty, -\frac{3}{2}) \cup (\frac{13}{2}, \infty) \).
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Absolute Value Functions
An absolute value function outputs the distance of a number from zero, always yielding a non-negative result. For example, |2x - 5| represents the distance of the expression (2x - 5) from zero, affecting how the function behaves on either side of the point where 2x - 5 equals zero.
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Function Composition
Inequalities Involving Absolute Values
Solving inequalities with absolute values requires considering two cases: one where the expression inside the absolute value is positive and one where it is negative. For y > 9 with y = |2x - 5| + 1, you isolate the absolute value and solve the resulting inequalities separately to find the solution set.
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06:07Linear Inequalities
Interval Notation
Interval notation is a concise way to represent sets of numbers between two endpoints. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints). This notation is essential for expressing the solution set of inequalities clearly and efficiently.
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05:18Interval Notation
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