Domanda del libro di testo
Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x2 but with the given point as the vertex. (5, 3)
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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 49
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. (4−2x)/(3x+4)≤0
Guida verificata passo dopo passo1
Identify the rational inequality: \(\frac{4 - 2x}{3x + 4} \leq 0\).
Find the critical points by setting the numerator and denominator equal to zero separately: solve \(4 - 2x = 0\) and \(3x + 4 = 0\).
Determine the values of \(x\) where the expression is undefined (denominator zero) and where the expression equals zero (numerator zero). These points divide the number line into intervals.
Test a value from each interval in the inequality \(\frac{4 - 2x}{3x + 4} \leq 0\) to check whether the expression is negative or zero in that interval.
Combine the intervals where the inequality holds true, and express the solution set in interval notation, remembering to exclude points where the denominator is zero.
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Rational Inequalities
Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.
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3:21
Nonlinear Inequalities
Critical Points and Sign Analysis
Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.
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05:46Point-Slope Form
Interval Notation and Graphing Solutions
After determining solution intervals, express them using interval notation, which concisely represents sets of real numbers. Graphing on a number line visually shows these intervals, indicating included or excluded endpoints based on inequality type.
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05:18Interval Notation
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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. (3x+5)/(6−2x)≥0
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Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x+1) − 2
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In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors.
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