Domanda del libro di testo
In Exercises 19–28, solve each system by the addition method.
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 6, Problema 23
Graph each inequality. y>2x
Guida verificata passo dopo passo1
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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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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6:13
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.
Video consigliato:
7:02Linear Inequalities
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.
Video consigliato:
05:10Graphs & the Rectangular Coordinate System
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