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

Verified step by step guidance

1. Recognize that the expression inside the absolute value is \(x^4 + 2x^2 + 1\).
2. Understand that the absolute value of any expression is always non-negative, meaning it is \(\geq 0\).
3. Since the inequality is \(|x^4 + 2x^2 + 1| < 0\), note that this is asking for when the absolute value is negative.
4. Conclude that there are no real solutions because an absolute value cannot be less than zero.
5. Verify by considering the properties of absolute values and the non-negativity of \(x^4 + 2x^2 + 1\).

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.