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5. Solve the inequality 12−x+52+x<1\frac{1}{2-x}+\frac{5}{2+x}<12−x1+2+x5<1.
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# Solution:
12−x+52+x<1,2+x+5(2−x)−1(2−x)(2+x)(2−x)(2+x)<0\frac{1}{2-x}+\frac{5}{2+x}<1, \frac{2+x+5(2-x)-1(2-x)(2+x)}{(2-x)(2+x)}<02−x1+2+x5<1,(2−x)(2+x)2+x+5(2−x)−1(2−x)(2+x)<0,
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2+x+10−5x+x2−4(2−x)(2+x)<0,x2−4x+8(2−x)(2+x)<0\frac{2+x+10-5x+x^2-4}{(2-x)(2+x)}<0, \frac{x^2-4x+8}{(2-x)(2+x)}<0(2−x)(2+x)2+x+10−5x+x2−4<0,(2−x)(2+x)x2−4x+8<0
Answer: (−∞,−2)∪(2,+∞)(-\infty, -2) \cup (2, +\infty)(−∞,−2)∪(2,+∞).
Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.