Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | x^4 + 2x^2 + 1 | < 0

Key Concepts

Absolute Value Properties

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any expression |A|, it is true that |A| ≥ 0. This property is crucial when solving inequalities involving absolute values, as it helps determine the conditions under which the expression can be less than zero.

Quadratic Expressions

The expression x^4 + 2x^2 + 1 can be viewed as a quadratic in terms of x^2. By substituting y = x^2, the expression transforms into y^2 + 2y + 1, which factors to (y + 1)^2. Understanding how to manipulate and factor quadratic expressions is essential for solving the given inequality.

Inequalities and Their Solutions

Inequalities express a relationship where one side is not equal to the other, often involving greater than or less than symbols. To solve an inequality like |A| < 0, one must recognize that it implies A must be negative, which is impossible for absolute values. Thus, understanding the implications of inequalities is key to determining the solution set.