Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Inequalities

College Algebra

1. Equations & Inequalities

Linear Inequalities

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1:33 minutes

Problem 24

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. −x^2+2x≥0

Verified step by step guidance

Rewrite the inequality in standard form: \(-x^2 + 2x \geq 0\).

Factor the quadratic expression: \(-x(x - 2) \geq 0\).

Identify the critical points by setting each factor equal to zero: \(-x = 0\) and \(x - 2 = 0\).

Solve for the critical points: \(x = 0\) and \(x = 2\).

Test intervals determined by the critical points \((x < 0, 0 < x < 2, x > 2)\) to determine where the inequality holds true.

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Key Concepts

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

Polynomial Inequalities

Polynomial inequalities involve expressions that compare a polynomial to a value, typically zero, using inequality symbols such as ≥, ≤, >, or <. To solve these inequalities, one must find the values of the variable that make the inequality true, often by determining the roots of the polynomial and testing intervals between these roots.

Graphing on a Number Line

Graphing the solution set of an inequality on a number line visually represents the range of values that satisfy the inequality. Solutions are typically indicated with open or closed circles to denote whether endpoints are included (closed) or excluded (open), and shaded regions show the intervals of valid solutions.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints.