Domanda del libro di testo
Find the inverse of f(x) = x3 + 2
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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 60
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/(x + 2) ≥ 2
Guida verificata passo dopo passo1
Start by rewriting the inequality: \(\frac{x}{x + 2} \geq 2\).
Bring all terms to one side to have zero on the other side: \(\frac{x}{x + 2} - 2 \geq 0\).
Find a common denominator and combine the terms: \(\frac{x - 2(x + 2)}{x + 2} \geq 0\).
Simplify the numerator: \(\frac{x - 2x - 4}{x + 2} = \frac{-x - 4}{x + 2} \geq 0\).
Determine the critical points by setting numerator and denominator equal to zero: numerator \(-x - 4 = 0\) and denominator \(x + 2 = 0\). These points divide the number line into intervals to test for the inequality.
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Rational Inequalities
Rational inequalities involve expressions where one rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.
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3:21
Nonlinear Inequalities
Critical Points and Sign Analysis
Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine where the rational expression is positive or negative, which helps identify the solution set for the inequality.
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05:46Point-Slope Form
Interval Notation and Graphing on a Number Line
Interval notation concisely represents sets of real numbers that satisfy the inequality, using parentheses or brackets to indicate open or closed intervals. Graphing these intervals on a number line visually shows the solution set and helps verify the correctness of the solution.
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05:18Interval Notation
Pratica correlata
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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)=3x5+2x4−15x3−10x2+12x+8
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Domanda del libro di testo
Find the domain of each function.
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Domanda del libro di testo
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. r(x) = (x^2 + 4x + 3)/(x + 2)^2
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Domanda del libro di testo
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 - 2)/(x + 2) ≤ 2
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Domanda del libro di testo
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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)
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