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Basics & Logarithms: Find Number Solutions

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

Question
Find the number of solutions of ∣x−6∣+∣x+6∣<12|x-6| + |x+6| < 12 is
Solution
Answer: 0
Step 1: By the triangle inequality:
∣x−6∣+∣x+6∣=∣(x−6)−(−(x+6))∣ at minimum=∣12∣=12|x-6| + |x+6| = |(x-6) - (-(x+6))| \text{ at minimum} = |12| = 12
 More precisely, ∣x−6∣+∣x+6∣|x-6|+|x+6| represents the sum of distances from xx to 66 and from xx to −6-6. This sum is always at least the distance between 66 and −6-6, which equals 1212. Step 2: Therefore ∣x−6∣+∣x+6∣≥12|x-6|+|x+6| \geq 12 for all real xx, with equality for −6≤x≤6-6 \leq x \leq 6. The strict inequality <12< 12 has no solution. Answer: 0
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