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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 27
Chapter 6, Problem 27

Graph each inequality. y > 2x + 1

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1
Identify the boundary curve by rewriting the inequality as an equation: \(y = 2^{x} + 1\). This curve will help us determine the region to shade.
Graph the function \(y = 2^{x} + 1\). Since \$2^{x}\( is an exponential function, the graph will rise rapidly as \)x$ increases, and the entire graph will be shifted up by 1 unit.
Because the inequality is strict (\(y > 2^{x} + 1\)), draw the boundary curve as a dashed line to indicate that points on the curve are not included in the solution set.
Choose a test point not on the boundary curve, such as \((0,0)\), and substitute it into the inequality \(y > 2^{x} + 1\) to check if it satisfies the inequality. If it does, shade the region containing that point; if not, shade the opposite side.
Shade the region above the dashed curve \(y = 2^{x} + 1\) to represent all points where \(y\) is greater than \(2^{x} + 1\), completing the graph of the inequality.

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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 variable is in the exponent. In this question, y = 2^x + 1 shifts the basic exponential graph y = 2^x upward by 1 unit. Understanding the shape and behavior of exponential functions is essential for graphing and interpreting inequalities involving them.
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Graphing Inequalities

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For y > 2^x + 1, you first graph the boundary curve y = 2^x + 1, then shade the area above this curve because y is greater than the function values. The boundary is usually drawn as a dashed line when the inequality is strict (>) to indicate points on the line are not included.
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Linear Inequalities

Transformations of Functions

Transformations modify the graph of a function by shifting, stretching, or reflecting it. The '+1' in y = 2^x + 1 shifts the graph of y = 2^x vertically upward by 1 unit. Recognizing such transformations helps in accurately plotting the function and understanding how the inequality's boundary changes.
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Related Practice
Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

x2+2y2=9 x^2 + 2y^2 = 9

x2+y2=25x^2+y^2=25

Textbook Question

Graph each inequality. y ≤ log(x - 1) - 2

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Textbook Question

Solve each system, using the method indicated.

3x + y = -7

x - y = -5 (Gaussian elimination)

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Textbook Question

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

2x - y = 6

4x - 2y = 0

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Textbook Question

Find the inverse, if it exists, for each matrix.[22−4260−3−35]\(\left\)[ \(\begin{matrix}\) 2 & 2 & -4 \\2 & 6 & 0 \\-3 & -3 & 5 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. 1/(x(2x + 1)(3x2 + 4))