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Home » We test small values of $ n $: - Groen Casting

We test small values of $ n $: - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 #### 96Question: What is the sum of all values of $ x $ for which $ \sqrt{(x - 3)^2} = 2x - 5 $?

📌 Solution: The equation $ \sqrt{(x - 3)^2} = 2x - 5 $ simplifies to $ |x - 3| = 2x - 5 $. This gives two cases:

📌 $ x - 3 = 2x - 5 $ → $ -3 + 5 = 2x - x $ → $ x = 2 $.

📌 $ -(x - 3) = 2x - 5 $ → $ -x + 3 = 2x - 5 $ → $ 3 + 5 = 3x $ → $ x = \frac{8}{3} $.

📌 Check validity: For $ x = 2 $, $ |2 - 3| = 1 $ and $ 2(2) - 5 = -1 $. Not valid since LHS ≠ RHS. For $ x = \frac{8}{3} $, $ \left|\frac{8}{3} - 3\right| = \frac{1}{3} $ and $ 2\left(\frac{8}{3}\right) - 5 = \frac{1}{3} $. Valid.

📌 Only $ x = \frac{8}{3} $ is valid. The sum is $ \boxed{\dfrac{8}{3}} $.

📌 Question: How many positive 3-digit numbers are divisible by 7 but not by 5?

📌 Solution: First, find 3-digit numbers divisible by 7: The smallest is 105 ($ 7 \times 15 $), the largest is 994 ($ 7 \times 142 $). Total numbers: $ 142 - 14 = 128 $.

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