Textbook Question
Write each complex number in standard form. (2 + 3i)3
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 2, Problem 64
In Exercises 59–94, solve each absolute value inequality. |3x + 5| < 17
Verified step by step guidance1
Recall that an absolute value inequality of the form \(|A| < B\) (where \(B > 0\)) can be rewritten as a compound inequality: \(-B < A < B\).
Apply this rule to the given inequality \(|3x + 5| < 17\), which becomes \(-17 < 3x + 5 < 17\).
Next, solve the compound inequality by isolating \(x\). Start by subtracting 5 from all parts: \(-17 - 5 < 3x < 17 - 5\).
Simplify the inequalities: \(-22 < 3x < 12\).
Finally, divide all parts by 3 to solve for \(x\): \(\frac{-22}{3} < x < 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Definition
The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in rewriting absolute value inequalities into equivalent compound inequalities.
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Vertex Form
Solving Absolute Value Inequalities
An inequality of the form |A| < B (where B > 0) can be rewritten as a compound inequality: -B < A < B. This means the expression inside the absolute value lies between -B and B. This approach allows solving the inequality by breaking it into two linear inequalities.
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Linear Inequalities
Once the absolute value inequality is rewritten without absolute values, it becomes a linear inequality or a system of inequalities. Solving these involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division, while considering the direction of the inequality.
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Related Practice
Textbook Question
Solve each equation in Exercises 65–74 using the quadratic formula.
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Textbook Question
In Exercises 59–94, solve each absolute value inequality. |2x - 6| < 8
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Textbook Question
Solve each equation in Exercises 47–64 by completing the square. 3x2 - 5x - 10 = 0
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Textbook Question
In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |x - 2| = 7
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Textbook Question
Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.
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