Textbook Question
Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.
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Ch. 3 - Polynomial and Rational Functions
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 4, Problem 73
In Exercises 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) ≤ - 3/4(x - 2)
Verified step by step guidance1
Identify the inequality to solve: \(\frac{1}{4}(x + 2) \leq -\frac{3}{4}(x - 2)\).
Eliminate the fractions by multiplying both sides of the inequality by 4 to simplify the expression.
Distribute the multiplication on both sides: \(x + 2 \leq -3(x - 2)\).
Expand the right side: \(x + 2 \leq -3x + 6\).
Collect like terms by adding \$3x\( to both sides and subtracting 2 from both sides to isolate \)x$: \(x + 3x \leq 6 - 2\) which simplifies to \(4x \leq 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions and Their Graphs
A rational function is a ratio of two polynomials, and its graph can have vertical and horizontal asymptotes where the function is undefined or approaches a limit. Understanding the behavior near these asymptotes and intercepts helps analyze the function's values and solve inequalities involving the function.
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How to Graph Rational Functions
Solving Inequalities Involving Rational Expressions
To solve inequalities with rational expressions, identify critical points where the numerator or denominator is zero, then test intervals between these points to determine where the inequality holds. Graphs can visually aid in identifying solution intervals by showing where the function lies above or below a certain value.
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Guided course
02:58Rationalizing Denominators
Asymptotes and Domain Restrictions
Vertical asymptotes occur where the denominator is zero, indicating values excluded from the domain. Horizontal asymptotes describe end behavior as x approaches infinity. Recognizing these helps in understanding the function's domain and range, which is crucial when solving inequalities or interpreting graphs.
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Related Practice
Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5
Textbook Question
In Exercises 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) > - 3/4(x - 2)
Textbook Question
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
2x3 + 11x2 < 7x + 6
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Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)
Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x2+x−6)