Domanda del libro di testo
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. −x2 + x ≥ 0
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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 22
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 3x2 − 5x ≤ 0
Guida verificata passo dopo passo1
Start by writing the inequality: \(3x^2 - 5x \leq 0\).
Factor the left-hand side expression: factor out the common term \(x\) to get \(x(3x - 5) \leq 0\).
Identify the critical points by setting each factor equal to zero: \(x = 0\) and \(3x - 5 = 0\), which gives \(x = \frac{5}{3}\).
Determine the sign of the product \(x(3x - 5)\) in the intervals defined by the critical points: \((-\infty, 0)\), \((0, \frac{5}{3})\), and \((\frac{5}{3}, \infty)\).
Use the sign analysis to find where the product is less than or equal to zero, then express the solution set in interval notation and graph it on the real number line.
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Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (e.g., ≤, ≥, <, >). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.
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06:07
Linear Inequalities
Factoring Quadratic Expressions
Factoring is the process of rewriting a quadratic polynomial as a product of simpler binomials or monomials. For example, 3x² - 5x can be factored as x(3x - 5). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.
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06:08Solving Quadratic Equations by Factoring
Interval Notation and Number Line Graphing
Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive, negative, or zero. Together, they help communicate the solution set clearly and precisely.
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05:18Interval Notation
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