Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x−4)(x+2)>0
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Ch. 3 - Polynomial and Rational Functions
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 4, Problem 1
Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x. y = 65 when x = 5. Find y when x = 12.
Verified step by step guidance1
Identify the type of variation described. Since y varies directly as x, we can write the equation as \(y = kx\), where \(k\) is the constant of proportionality.
Use the given values to find the constant \(k\). Substitute \(y = 65\) and \(x = 5\) into the equation \(y = kx\) to get \(65 = k \times 5\).
Solve for \(k\) by dividing both sides of the equation by 5, resulting in \(k = \frac{65}{5}\).
Write the specific variation equation using the value of \(k\) found: \(y = kx\) becomes \(y = \left(\frac{65}{5}\right) x\).
Find \(y\) when \(x = 12\) by substituting \(x = 12\) into the equation \(y = \left(\frac{65}{5}\right) x\) and simplify to express \(y\) in terms of known values.
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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 variable is a constant multiple of another, expressed as y = kx. Here, y changes proportionally with x, meaning if x doubles, y also doubles. Understanding this helps set up the equation to find unknown values.
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Constant of Variation
The constant of variation (k) is the fixed multiplier linking x and y in direct variation. It is found by substituting known values of x and y into y = kx. Once k is determined, it can be used to find y for any given x.
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Four-Step Procedure for Variation Problems
This procedure involves: 1) identifying the type of variation, 2) writing the variation equation, 3) finding the constant of variation using given values, and 4) using the equation to find the unknown variable. It provides a systematic approach to solve variation problems.
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Related Practice
Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)(x−5)>0
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Textbook Question
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Textbook Question
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Textbook Question
In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range.
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