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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−4∣+3∣x+6∣−2x=5|x-4| + 3|x+6| - 2x = 5 is
Solution
Answer: 0
Critical points: x=−6x = -6 and x=4x = 4. Case 1: x<−6x < -6: (4−x)+3(−x−6)−2x=5  ⟹  −6x−14=5  ⟹  x=−196≈−3.2(4-x) + 3(-x-6) - 2x = 5 \implies -6x - 14 = 5 \implies x = -\dfrac{19}{6} \approx -3.2 Not in x<−6x < -6. Rejected. Case 2: −6≤x≤4-6 \leq x \leq 4: (4−x)+3(x+6)−2x=5  ⟹  22=5(4-x) + 3(x+6) - 2x = 5 \implies 22 = 5 Contradiction. No solution in this interval. Case 3: x>4x > 4: (x−4)+3(x+6)−2x=5  ⟹  2x+14=5  ⟹  x=−92(x-4) + 3(x+6) - 2x = 5 \implies 2x + 14 = 5 \implies x = -\dfrac{9}{2} Not in x>4x > 4. Rejected. Answer: 0
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