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Home » Case 1: \( 2x - 5 = x + 3 \Rightarrow x = 8 \) - ClickBalance

Case 1: \( 2x - 5 = x + 3 \Rightarrow x = 8 \) - ClickBalance

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Set equal to 8:

📌 Hence, the truth-weight parameter is $ oxed{6} $.Question: Find $ x $ such that the vectors $ egin{pmatrix} x \ 2 \end{pmatrix} $ and $ egin{pmatrix} 3 \ -x \end{pmatrix} $ are orthogonal.

📌 Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: $ x \cdot 3 + 2 \cdot (-x) = 3x - 2x = x $. Set $ x = 0 $. Thus, $ oxed{0} $ is the solution.

📌 Question: Determine the number of real solutions to the equation $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $.

📌 Solution: Let $ y = \sin(2x) $. The equation becomes $ y^2 + 3y + 2 = 0 $, which factors as $ (y + 1)(y + 2) = 0 $. Thus, $ y = -1 $ or $ y = -2 $. Since $ \sin(2x) $ ranges between $-1$ and $1$, $ y = -2 $ is invalid. For $ y = -1 $, $ \sin(2x) = -1 $ has infinitely many solutions (e.g., $ 2x = rac{3\pi}{2} + 2\pi k $, $ k \in \mathbb{Z} $). However, if restricted to a specific interval (not stated here), the count would depend on the domain. Assuming $ x \in \mathbb{R} $, there are infinitely many solutions. But if the question implies a general count, the answer is oxed{ ext{infinitely many}}.

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