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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblema 93a
Capitolo 2, Problema 93a

Solve each absolute value inequality. 4 + |3 - x/3| ≥ 9

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1
Rewrite the inequality by isolating the absolute value expression. Subtract 4 from both sides: |3 - x/3| ≥ 5.
Understand the definition of absolute value inequalities. For |A| ≥ B (where B > 0), this means A ≤ -B or A ≥ B.
Apply the definition to the inequality |3 - x/3| ≥ 5. This splits into two cases: (1) 3 - x/3 ≤ -5 and (2) 3 - x/3 ≥ 5.
Solve the first case (3 - x/3 ≤ -5): Subtract 3 from both sides to get -x/3 ≤ -8. Then multiply through by -3 (remember to reverse the inequality sign when multiplying by a negative number) to get x ≥ 24.
Solve the second case (3 - x/3 ≥ 5): Subtract 3 from both sides to get -x/3 ≥ 2. Then multiply through by -3 (again reversing the inequality sign) to get x ≤ -6. Combine the solutions: x ≤ -6 or x ≥ 24.

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Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value bars.
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Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving inequalities, especially those involving absolute values, it is important to consider the different cases that arise from the definition of absolute value, leading to multiple potential solutions.
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Solving Absolute Value Inequalities

To solve an absolute value inequality, one must isolate the absolute value expression and then split the inequality into two separate cases. For example, if |A| ≥ B, it leads to two scenarios: A ≥ B or A ≤ -B. This method allows for finding all possible solutions that satisfy the original inequality, which is essential for complete problem-solving.
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