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 74
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
Start by rewriting the inequality \(\frac{1}{4}(x + 2) > -\frac{3}{4}(x - 2)\) to have all terms on one side for easier manipulation.
Multiply both sides of the inequality by 4 to eliminate the denominators, resulting in \(x + 2 > -3(x - 2)\).
Distribute the \(-3\) on the right side to get \(x + 2 > -3x + 6\).
Add \$3x\( to both sides and subtract 2 from both sides to isolate \)x\( terms on one side: \)x + 3x > 6 - 2$ which simplifies to \(4x > 4\).
Divide both sides by 4 to solve for \(x\), giving \(x > 1\). This is the solution to the inequality.
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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, expressed as f(x) = P(x)/Q(x). Its graph can have vertical asymptotes where the denominator is zero and horizontal or oblique asymptotes depending on the degrees of numerator and denominator. Understanding these features helps analyze the behavior of the function and solve inequalities involving it.
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Solving Inequalities Involving Rational Expressions
To solve inequalities with rational expressions, first bring all terms to one side to form a single rational expression. Then determine where the numerator and denominator are zero to find critical points. Use these points to test intervals on the number line to see where the inequality holds true.
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Interpreting Graphs to Solve Inequalities
Graphs of rational functions show where the function is positive or negative by its position relative to the x-axis. Points where the graph crosses or touches the x-axis correspond to zeros of the numerator. Vertical asymptotes indicate values excluded from the domain. Using the graph, one can visually identify solution intervals for inequalities.
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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 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 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) ≤ - 3/4(x - 2)
Textbook Question
Follow the seven steps to graph each rational function. f(x)=x4/(x2+2)
Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)