Rewrite each expression without the absolute value bars. |√2-1|

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Recall that the absolute value of a number \(x\), denoted \(|x|\), is defined as \(x\) if \(x \geq 0\), and \(-x\) if \(x < 0\).

Identify the expression inside the absolute value bars: \(\sqrt{2} - 1\).

Determine whether \(\sqrt{2} - 1\) is nonnegative or negative by approximating \(\sqrt{2}\). Since \(\sqrt{2} \approx 1.414\), then \(\sqrt{2} - 1 \approx 0.414\), which is positive.

Since \(\sqrt{2} - 1 \geq 0\), the absolute value expression \(|\sqrt{2} - 1|\) can be rewritten without the absolute value bars as \(\sqrt{2} - 1\).

Therefore, the expression without absolute value bars is simply \(\sqrt{2} - 1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always yielding a non-negative result. For any real number x, |x| equals x if x is non-negative, and -x if x is negative.

Evaluating Square Roots

The square root function, √x, returns the non-negative number whose square is x. Since √2 is approximately 1.414, it is positive, which helps determine the sign of expressions involving square roots.

Simplifying Expressions Inside Absolute Value

To rewrite an expression without absolute value bars, first evaluate or estimate the expression inside. If the expression is positive or zero, the absolute value can be removed directly; if negative, multiply by -1 to remove the bars.