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Basics & Logarithms: Find Number Single Digit Negative Integers Satisfying

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

Question
Find the number of single digit negative integers satisfying ∣∣x−6∣−∣x+6∣∣=12\big||x-6| - |x+6|\big| = 12 is
Solution
Answer: 4
Step 1: Write a=x−6a = x-6 and b=x+6b = x+6. Note a−b=−12a - b = -12, so ∣a−b∣=12|a-b| = 12. Step 2: The reverse triangle inequality gives ∣∣a∣−∣b∣∣≤∣a−b∣=12\big||a|-|b|\big| \leq |a-b| = 12. Equality ∣∣a∣−∣b∣∣=∣a−b∣\big||a|-|b|\big| = |a-b| holds if and only if ab≥0ab \geq 0. Step 3: (x−6)(x+6)≥0  ⟹  x2≥36  ⟹  ∣x∣≥6(x-6)(x+6) \geq 0 \implies x^2 \geq 36 \implies |x| \geq 6, i.e., x≤−6x \leq -6 or x≥6x \geq 6. Step 4: Single digit negative integers satisfying x≤−6x \leq -6: {−6,−7,−8,−9}\{-6, -7, -8, -9\}. Count =4= 4. Answer: 4
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