![1)\frac{1}{x}<1\\\\\frac{1}{x}-1<0\\\\\frac{1-x}{x}<0\\\\x(x-1)>0](https://tex.z-dn.net/?f=1%29%5Cfrac%7B1%7D%7Bx%7D%3C1%5C%5C%5C%5C%5Cfrac%7B1%7D%7Bx%7D-1%3C0%5C%5C%5C%5C%5Cfrac%7B1-x%7D%7Bx%7D%3C0%5C%5C%5C%5Cx%28x-1%29%3E0)
+ - +
_______₀_______₀_______
0 1
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Ответ : x ∈ (- ∞ ; 0 ) ∪ (1 ; + ∞)
![2)\frac{x}{x+3}>\frac{1}{2}\\\\\frac{x}{x+3}-\frac{1}{2}>0\\\\\frac{2x-x-3}{2(x+3)}>0\\\\\frac{x-3}{2(x+3)}>0\\\\(x-3)(x+3)>0](https://tex.z-dn.net/?f=2%29%5Cfrac%7Bx%7D%7Bx%2B3%7D%3E%5Cfrac%7B1%7D%7B2%7D%5C%5C%5C%5C%5Cfrac%7Bx%7D%7Bx%2B3%7D-%5Cfrac%7B1%7D%7B2%7D%3E0%5C%5C%5C%5C%5Cfrac%7B2x-x-3%7D%7B2%28x%2B3%29%7D%3E0%5C%5C%5C%5C%5Cfrac%7Bx-3%7D%7B2%28x%2B3%29%7D%3E0%5C%5C%5C%5C%28x-3%29%28x%2B3%29%3E0)
+ - +
_______₀________₀_________
- 3 3
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Ответ : x ∈ (- ∞ ; - 3) ∪ (3 ; + ∞)
![3)\frac{1}{x+2}<\frac{3}{x-3}\\\\\frac{1}{x+2}-\frac{3}{x-3}<0\\\\\frac{x-3-3x-6}{(x+2)(x-3)}<0\\\\\frac{-2x-9}{(x+2)(x-3)}<0\\\\(x+4,5)(x+2)(x-3)>0](https://tex.z-dn.net/?f=3%29%5Cfrac%7B1%7D%7Bx%2B2%7D%3C%5Cfrac%7B3%7D%7Bx-3%7D%5C%5C%5C%5C%5Cfrac%7B1%7D%7Bx%2B2%7D-%5Cfrac%7B3%7D%7Bx-3%7D%3C0%5C%5C%5C%5C%5Cfrac%7Bx-3-3x-6%7D%7B%28x%2B2%29%28x-3%29%7D%3C0%5C%5C%5C%5C%5Cfrac%7B-2x-9%7D%7B%28x%2B2%29%28x-3%29%7D%3C0%5C%5C%5C%5C%28x%2B4%2C5%29%28x%2B2%29%28x-3%29%3E0)
- + - +
_______₀________₀________₀________
- 4,5 - 2 3
Ответ : x ∈ (- 4,5 ; - 2) ∪ (3 ; + ∞)
![4)\frac{4}{x+1}+\frac{2}{1-x} <1\\\\\frac{4}{x+1}+\frac{2}{1-x}-1<0\\\\\frac{4-4x+2x+2-1+x^{2}}{(x+1)(1-x)}<0\\\\\frac{x^{2}-2x+5}{(x+1)(1-x)}<0\\\\\ \frac{x^{2}-2x+5 }{(x+1)(x-1)}>0](https://tex.z-dn.net/?f=4%29%5Cfrac%7B4%7D%7Bx%2B1%7D%2B%5Cfrac%7B2%7D%7B1-x%7D%20%3C1%5C%5C%5C%5C%5Cfrac%7B4%7D%7Bx%2B1%7D%2B%5Cfrac%7B2%7D%7B1-x%7D-1%3C0%5C%5C%5C%5C%5Cfrac%7B4-4x%2B2x%2B2-1%2Bx%5E%7B2%7D%7D%7B%28x%2B1%29%281-x%29%7D%3C0%5C%5C%5C%5C%5Cfrac%7Bx%5E%7B2%7D-2x%2B5%7D%7B%28x%2B1%29%281-x%29%7D%3C0%5C%5C%5C%5C%5C%20%5Cfrac%7Bx%5E%7B2%7D-2x%2B5%20%7D%7B%28x%2B1%29%28x-1%29%7D%3E0)
x² - 2x + 5 > 0 при любых действительных значениях x , значит достаточно решить неравенство :
(x + 1)(x - 1) > 0
+ - +
________₀_______₀_______
- 1 1
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Ответ : x ∈ (- ∞ ; - 1) ∪ (1 ; + ∞)