Textbook Question
Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)
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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 65
Write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is at most 4. The y-variable added to the product of 3 and the x-variable does not exceed 6.
Verified step by step guidance1
Identify the variables: let the x-variable be \(x\) and the y-variable be \(y\).
Translate the first sentence "The sum of the x-variable and the y-variable is at most 4" into the inequality: \(x + y \leq 4\).
Translate the second sentence "The y-variable added to the product of 3 and the x-variable does not exceed 6" into the inequality: \(y + 3x \leq 6\).
Write the system of inequalities as:
\[
\begin{cases}
x + y \leq 4 \\
y + 3x \leq 6
\end{cases}
\]
To graph the system, first graph the boundary lines \(x + y = 4\) and \(y + 3x = 6\) by finding intercepts or using slope-intercept form, then shade the regions that satisfy each inequality (below or on the lines). The solution to the system is the overlapping shaded region.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Translating Word Problems into Inequalities
This involves converting verbal statements into mathematical inequalities. Key phrases like 'at most' or 'does not exceed' indicate 'less than or equal to' (≤). Identifying variables and their relationships is essential to form accurate inequalities representing the problem.
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Linear Inequalities
Systems of Inequalities
A system of inequalities consists of two or more inequalities with the same variables. Solutions must satisfy all inequalities simultaneously. Understanding how to write and interpret these systems is crucial for analyzing constraints in problems.
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Graphing Inequalities in Two Variables
Graphing inequalities involves shading regions on the coordinate plane that satisfy the inequality. The boundary line is drawn using equality, solid if ≤ or ≥, dashed if < or >. The solution to a system is the overlapping shaded region of all inequalities.
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Related Practice
Textbook Question
In Exercises 69–70, rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. |x|≤2, |y|≤3
Textbook Question
In Exercises 65–68, write the given sentences as a system of inequalities in two variables. Then graph the system. The sum of the x-variable and the y-variable is no more than 2. The y-variabe is no less than the difference between the square of the x-variable and 4.
Textbook Question
In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 3x+y≤6, 2x−y≤−1, x>−2, y<4
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Textbook Question
Use a system of linear equations to solve Exercises 73–84. How many ounces of a 15% alcohol solution must be mixed with 4 ounces of a 20% alcohol solution to make a 17% alcohol solution?
Textbook Question
In Exercises 63–64, write each sentence as an inequality in two variables. Then graph the inequality. The y-variable is at least 4 more than the product of -2 and the x-variable.