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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 19
Capitolo 4, Problema 19

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. x2−4x≥0x^2−4x≥0

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Start by rewriting the inequality: \(x^2 - 4x \geq 0\).
Factor the left-hand side expression: \(x^2 - 4x = x(x - 4)\).
Identify the critical points by setting each factor equal to zero: \(x = 0\) and \(x - 4 = 0 \Rightarrow x = 4\).
Determine the sign of the product \(x(x - 4)\) in the intervals defined by the critical points: \((-\infty, 0)\), \((0, 4)\), and \((4, \infty)\).
Write the solution set by including the intervals where the product is greater than or equal to zero, and express it in interval notation.

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Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). 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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Linear Inequalities

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic expression as a product of simpler polynomials. For example, x² - 4x can be factored as x(x - 4). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.
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Solving 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 satisfies the inequality, using open or closed dots to indicate whether endpoints are included, corresponding to strict or inclusive inequalities.
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Interval Notation
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