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)
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 72
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
2x3 + 11x2 < 7x + 6
Verified step by step guidance1
Rewrite the given inequality \(2x^{3} + 11x^{2} < 7x + 6\) by bringing all terms to one side to set the inequality to zero: \(2x^{3} + 11x^{2} - 7x - 6 < 0\).
Recognize that the left side of the inequality is the polynomial function \(f(x) = 2x^{3} + 11x^{2} - 7x - 6\) whose graph is provided.
Identify the x-intercepts (roots) of the polynomial from the graph, which are approximately at \(x = -7\), \(x = -3\), and \(x = 1\). These points divide the x-axis into intervals.
Determine the sign of \(f(x)\) on each interval by observing whether the graph is above or below the x-axis: where the graph is below the x-axis, \(f(x) < 0\); where it is above, \(f(x) > 0\).
Write the solution to the inequality \(f(x) < 0\) as the union of intervals where the graph lies below the x-axis, using the roots as boundaries.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality signs. Solving them requires determining where the polynomial function is greater than or less than a given value, often by analyzing the sign of the polynomial over intervals.
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Linear Inequalities
Graph Interpretation of Polynomial Functions
The graph of a polynomial function shows its behavior, including where it crosses the x-axis (roots) and where it is positive or negative. By examining the graph, one can identify intervals where the function lies above or below a certain value, which helps solve inequalities.
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05:25Graphing Polynomial Functions
Finding Roots and Critical Points
Roots are the x-values where the polynomial equals zero, marking boundary points for inequality solutions. Critical points, where the function changes direction, help determine the shape of the graph and the sign of the polynomial in different intervals.
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02:20Imaginary Roots with the Square Root Property
Related Practice
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.
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Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 9x + 4 ≥ 0
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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)