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Home » how to make city water better with out filter system - inBeat

how to make city water better with out filter system - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Thus, \(\mathbf{v} = \langle 1, t, 1 \rangle\) for any real \(t\).

📌 \boxed{\langle 1, t, 1 \rangle \text{ for any } t \in \mathbb{R}}

📌 Find the range of the function \( f(x) = \frac{x^2 + 1}{x^2 - 1} \) as \( x \) ranges over all real numbers such that \( x \neq \pm 1 \).

📌 Let \( y = \frac{x^2 + 1}{x^2 - 1} \). Rearranging gives:

📌 y(x^2 - 1) = x^2 + 1 \quad \Rightarrow \quad yx^2 - y = x^2 + 1.

📌 Rearranging terms:

📌 If \( y \neq 1 \),

📌 x^2 = \frac{y + 1}{y - 1}.

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