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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 4, Problema 58
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation.1/(x - 3) < 1
Guida verificata passo dopo passo1
Start by rewriting the inequality: \(\frac{1}{x - 3} < 1\).
Bring all terms to one side to have zero on the other side: \(\frac{1}{x - 3} - 1 < 0\).
Find a common denominator and combine the terms: \(\frac{1 - (x - 3)}{x - 3} < 0\).
Simplify the numerator: \(\frac{1 - x + 3}{x - 3} < 0\), which becomes \(\frac{4 - x}{x - 3} < 0\).
Determine the critical points by setting numerator and denominator equal to zero: numerator \(4 - x = 0\) gives \(x = 4\), denominator \(x - 3 = 0\) gives \(x = 3\). Use these points to test intervals on the number line and find where the inequality holds true.
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Rational Inequalities
Rational inequalities involve expressions where one or both sides contain rational functions, which are ratios of polynomials. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the expression over different intervals.
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Nonlinear Inequalities
Critical Points and Sign Analysis
Critical points are values where the rational expression is zero or undefined, such as where the numerator or denominator equals zero. These points divide the number line into intervals, and testing each interval helps determine where the inequality holds true.
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05:46Point-Slope Form
Interval Notation and Graphing Solutions
Interval notation is a concise way to express solution sets using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually represents these intervals, showing where the inequality is satisfied and highlighting excluded points.
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05:18Interval Notation
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In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)
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Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=2x4−3x3−7x2−8x+6
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Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x + 4)/(2x - 1) ≤ 3
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In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (2x - 4)/(x + 3)
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Follow the seven steps to graph each rational function. f(x)=2x/(x2−4)
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