Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 19
Chapter 4, Problem 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

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
06:07
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.
Recommended video:
06:08
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.
Recommended video:
05:18
Interval Notation
Related Practice
Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=x3−x2−9x+9f(x) = x^3 - x^2 - 9x + 9

Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x3+7x2−x+9f(x)=5x^3+7x^2−x+9

1
views
Textbook Question

Divide using synthetic division. (3x2+7x−20)÷(x+5)

1
views
Textbook Question

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. 4x2 + 1 ≥ 4x

5
views
Textbook Question

Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.

2
views
Textbook Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−1)2+2