Textbook Question
In Exercises 16–24, write the partial fraction decomposition of each rational expression. (4x^3 + 5x^2 + 7x - 1)/(x^2 + x + 1)^2
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Ch. 5 - Systems of 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 6, Problem 23
Graph each inequality. y>2x
Verified step by step guidance1
Identify the inequality to graph: \(y > 2^{x}\). This means we want to graph all points where the \(y\)-value is greater than \$2^{x}$.
First, graph the boundary curve \(y = 2^{x}\). This is an exponential function where the base is 2, so it passes through points like \((0,1)\) since \(2^{0} = 1\), and increases as \(x\) increases.
Since the inequality is strict (\(>\)), draw the curve \(y = 2^{x}\) as a dashed line to indicate that points on the line are not included in the solution.
Determine which side of the curve to shade by testing a point not on the curve, such as \((0,0)\). Substitute into the inequality: \(0 > 2^{0}\) becomes \(0 > 1\), which is false, so do not shade below the curve.
Shade the region above the curve \(y = 2^{x}\), representing all points where \(y\) is greater than \$2^{x}$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
An exponential function has the form y = a^x, where the base a is a positive constant. In this question, y = 2^x represents an exponential growth function, which increases rapidly as x increases. Understanding its shape and behavior is essential for graphing related inequalities.
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Exponential Functions
Graphing Inequalities
Graphing an inequality like y > 2^x involves shading the region of the coordinate plane where the inequality holds true. The boundary curve y = 2^x is graphed first, and since the inequality is strict (greater than), the boundary is drawn as a dashed line to indicate points on the curve are not included.
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Coordinate Plane and Regions
The coordinate plane is divided by the graph of the function into regions. For y > 2^x, the region above the curve is shaded. Understanding how to identify and shade the correct region based on the inequality symbol is crucial for accurately representing the solution set.
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Related Practice
Textbook Question
In Exercises 19–28, solve each system by the addition method.
Textbook Question
Write the partial fraction decomposition of each rational expression. (6x-11)/(x − 1)²
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Textbook Question
In Exercises 16–24, write the partial fraction decomposition of each rational expression. (7x^2 - 7x + 23)/(x - 3)(x^2 + 4)
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Textbook Question
In Exercises 19–30, solve each system by the addition method. x + 2y = 2 - 4x + 3y = 25
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Textbook Question
Write the partial fraction decomposition of each rational expression. (x2-6x+3)/(x − 2)3