Textbook Question
In Exercises 39–45, graph each inequality. y ≤ x^2 - 1
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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 43
Graph the solution set of each system of inequalities or indicate that the system has no solution.
Verified step by step guidance1
Start by examining each inequality separately. The system is:
\[\begin{cases} 2x - y > 4 \\ 2x - y > -5 \end{cases}\]
Rewrite each inequality in terms of \(y\) to better understand the boundary lines:
For the first inequality:
\[2x - y > 4 \implies -y > 4 - 2x \implies y < 2x - 4\]
For the second inequality:
\[2x - y > -5 \implies -y > -5 - 2x \implies y < 2x + 5\]
Graph the boundary lines \(y = 2x - 4\) and \(y = 2x + 5\). Since the inequalities are strict (greater than), these lines will be dashed to indicate points on the line are not included.
Determine the solution region for each inequality:
- For \(y < 2x - 4\), shade the region below the line \(y = 2x - 4\).
- For \(y < 2x + 5\), shade the region below the line \(y = 2x + 5\).
The solution to the system is the intersection of these two shaded regions. Since \(2x - 4\) is always less than \(2x + 5\) for all \(x\), the region below \(y = 2x - 4\) is contained within the region below \(y = 2x + 5\). Therefore, the solution set is all points below the line \(y = 2x - 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Linear Inequalities
Graphing linear inequalities involves plotting the boundary line given by the corresponding linear equation and then shading the region that satisfies the inequality. The boundary line is dashed if the inequality is strict (>, <) and solid if it includes equality (≥, ≤). This visual representation helps identify all points that satisfy the inequality.
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Linear Inequalities
System of Inequalities
A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions satisfying each inequality. Graphing each inequality and finding the overlapping shaded region reveals the solution set, or shows if no solution exists when regions do not overlap.
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Slope-Intercept Form and Boundary Lines
Rearranging inequalities into slope-intercept form (y = mx + b) helps graph the boundary lines easily. For example, 2x - y > 4 can be rewritten as y < 2x - 4. Understanding the slope and y-intercept allows accurate drawing of the boundary line, which is essential for correctly shading the solution region.
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Related Practice
Textbook Question
In Exercises 39–45, graph each inequality. y ≤ 2^x
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Textbook Question
In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.
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Textbook Question
In Exercises 43–46, let x represent one number and let y represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is 7. If one number is subtracted from the other, their difference is -1. Find the numbers.
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Textbook Question
Perform each long division and write the partial fraction decomposition of the remainder term. (x5+2)/(x2-1)
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Textbook Question
Write the partial fraction decomposition of each rational expression. (4x2+3x+14)/(x3 - 8)
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