Question

Solve each inequality algebraically. $$x^{3}-9 x \leq 0$$

   Solve each inequality algebraically.
$$x^{3}-9 x \leq 0$$

Algebra and Trigonometry

Algebra and Trigonometry

Michael Sullivan 10th Edition

Chapter 5, Problem 59

Instant Answer

Solved by Expert Michelle Nguyen

06/18/2020

Step 1

We can factor out an $x$ to get $x(x^{2}-9) \leq 0$. Show more…

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Solve each inequality algebraically. $$x^{3}-9 x \leq 0$$

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Key Concepts

Sign Analysis

Sign analysis, often performed using a sign chart, is a method used to test the sign of each factor over the intervals defined by the roots. By determining whether the product of the factors is positive or negative within each interval, one can identify the solution set of the inequality.

Zero Product Property

The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. This principle helps in identifying the critical points or roots of the polynomial, which are essential boundaries for the inequality.

Factorization

Factorization is the process of rewriting a polynomial as a product of its simpler polynomial factors. This step is critical as it breaks down a complex inequality into simpler components, making it easier to determine where the expression is non-positive or non-negative.

Polynomial Inequalities

This concept involves solving inequalities where one or more polynomial expressions are compared to zero. The goal is to identify the intervals on the number line where the polynomial assumes positive, negative, or zero values.

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