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Basics & Logarithms: Find Number Positive Integers Satisfying

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

Question
Find the number of positive integers satisfying ∣x2−3x+2∣+∣x2+7x+12∣=10∣x+1∣|x^2-3x+2| + |x^2+7x+12| = 10|x+1| is
Solution
Answer: 2
Step 1: For positive integers, x>0x > 0, so x2+7x+12=(x+3)(x+4)>0x^2+7x+12 = (x+3)(x+4) > 0 and ∣x+1∣=x+1|x+1| = x+1. The equation becomes:
∣(x−1)(x−2)∣+x2+7x+12=10(x+1)=10x+10|(x-1)(x-2)| + x^2+7x+12 = 10(x+1) = 10x+10
∣(x−1)(x−2)∣=−x2+3x−2=−(x−1)(x−2)|(x-1)(x-2)| = -x^2 + 3x - 2 = -(x-1)(x-2)
 Step 2: ∣A∣=−A|A| = -A if and only if A≤0A \leq 0, so (x−1)(x−2)≤0(x-1)(x-2) \leq 0, i.e., 1≤x≤21 \leq x \leq 2. Positive integers in [1,2][1, 2]: {1,2}\{1, 2\}. Verify x=1x = 1: ∣0∣+20=20=10∣2∣|0|+20 = 20 = 10|2|. Confirmed. Verify x=2x = 2: ∣0∣+30=30=10∣3∣|0|+30 = 30 = 10|3|. Confirmed. Answer: 2
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