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.) | x² + 1 | - | 2x | = 0

Verified step by step guidance

1. Start by rewriting the equation: \(|x^2 + 1| - |2x| = 0\). This means \(|x^2 + 1| = |2x|\).
2. Recognize that \(x^2 + 1\) is always positive or zero because \(x^2 \geq 0\) and adding 1 makes it strictly positive. Therefore, \(|x^2 + 1| = x^2 + 1\) for all real \(x\).
3. Rewrite the equation without the absolute value on the first term: \(x^2 + 1 = |2x|\).
4. Express \(|2x|\) as \$2|x|\( to simplify: \)x^2 + 1 = 2|x|$.
5. To solve for \(x\), consider two cases based on the definition of absolute value: Case 1: \(x \geq 0\), then \(|x| = x\), so the equation becomes \(x^2 + 1 = 2x\). Case 2: \(x < 0\), then \(|x| = -x\), so the equation becomes \(x^2 + 1 = -2x\). Solve each quadratic equation separately to find the possible values of \(x\).

Key Concepts

Absolute Value Properties

Absolute value represents the distance of a number from zero on the number line, always non-negative. Key properties include |a| ≥ 0, |a| = |-a|, and |ab| = |a||b|. Understanding how to manipulate and simplify expressions involving absolute values is essential for solving equations like |x² + 1| - |2x| = 0.

Solving Absolute Value Equations

To solve equations involving absolute values, isolate the absolute value expressions and consider cases based on the definition of absolute value. For example, |A| = |B| implies A = B or A = -B. This approach helps break down complex equations into simpler ones that can be solved algebraically.

Quadratic Expressions and Their Properties

Quadratic expressions like x² + 1 are always non-negative since x² ≥ 0 and adding 1 keeps it positive. Recognizing this helps simplify absolute value expressions, as |x² + 1| = x² + 1. This insight reduces complexity when solving equations involving absolute values of quadratics.