Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 21
Write each formula as an English phrase using the word varies or proportional. C=2πr, where C is the circumference of a circle of radius r.
Verified step by step guidance1
Identify the variables in the formula: \(C\) represents the circumference of a circle, and \(r\) represents the radius of the circle.
Recognize the constant in the formula: \(2\pi\) is a constant multiplier, where \(\pi\) is approximately 3.14159.
Understand the relationship: Since \(C = 2\pi r\), the circumference \(C\) changes as the radius \(r\) changes, multiplied by the constant \(2\pi\).
Express the relationship using the word 'varies': We say that the circumference \(C\) varies directly as the radius \(r\).
Write the full English phrase: 'The circumference of a circle varies directly (or is proportional) to its radius, with the constant of proportionality being \(2\pi\).'
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Direct Variation
Direct variation describes a relationship where one quantity changes proportionally with another. If y varies directly as x, then y = kx for some constant k. In the given formula, circumference varies directly with the radius.
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Proportionality Constant
The proportionality constant is the fixed multiplier that relates two varying quantities. In the formula C = 2πr, the constant 2π shows how many times the radius is multiplied to get the circumference, indicating the strength of the proportional relationship.
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Translating Mathematical Formulas into Words
This involves expressing algebraic equations using clear English phrases. For example, 'C = 2πr' can be stated as 'The circumference varies directly as the radius, with 2π as the constant of proportionality.' This skill helps in understanding and communicating mathematical relationships.
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06:36Solving Quadratic Equations Using The Quadratic Formula
Related Practice
Textbook Question
Use synthetic division to perform each division. (x3 - 1) / (x-1)
Textbook Question
If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?
Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. 2x2 + 5 ≤ 11x
Textbook Question
Graph the following on the same coordinate system.
(a) y = (x - 2)2
(b) y = (x + 1)2
(c) y = (x + 3)2
(d) How do these graphs differ from the graph of y = x2?
Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.