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Basics & Logarithms: Find Number Single Digit Positive Integers Satisfying

JEE Maths question with a full step-by-step solution.

Question
Find the number of single digit positive integers satisfying ∣x−5∣+∣x+5∣=2∣x∣|x-5| + |x+5| = 2|x| is
Solution
Answer: 5
Step 1: Write a=x−5a = x-5 and b=x+5b = x+5. Then ∣a∣+∣b∣=2∣x∣=∣a+b∣|a| + |b| = 2|x| = |a+b|. The identity ∣a∣+∣b∣=∣a+b∣|a|+|b| = |a+b| holds if and only if ab≥0ab \geq 0. Step 2: (x−5)(x+5)≥0  ⟹  x2≥25  ⟹  ∣x∣≥5(x-5)(x+5) \geq 0 \implies x^2 \geq 25 \implies |x| \geq 5. Step 3: Single digit positive integers satisfying x≥5x \geq 5: {5,6,7,8,9}\{5, 6, 7, 8, 9\}. Count =5= 5. Answer: 5
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