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

검증된 단계별 안내

Start by rewriting the equation: $|x^2 + 1| - |2x| = 0$. This means $|x^2 + 1| = |2x|$.

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 $x$.

Rewrite the equation without the absolute value on the first term: $x^2 + 1 = |2x|$.

Express $|2x|$ as \$2|x|$ to simplify: $x^2 + 1 = 2|x|$.

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 $x$.

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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.

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