\sqrt{x-1}\\\\ODZ:\; \; \left \{ {{x\geq -2\; ,\; x\geq 1/3} \atop {x\geq 1}} \right. \; \; \to \; \; x\geq 1\\\\\underbrace {\sqrt{x+2}}_{\geq 0}>\underbrace {\sqrt{x-1}+\sqrt{3x-1}}_{\geq 0}\\\\x+2>(x-1)+2\cdot \sqrt{x-1}\cdot \sqrt{3x-1}+(3x-1)\\\\x+2>4x-2+2\cdot \sqrt{(x-1)(3x-1)}\\\\2\cdot \sqrt{(x-1)(3x-1)}<-3x+4\\\\4(x-1)(3x-1)<9x^2-24x+16\\\\4(3x^2-4x+1)<9x^2-24x+16\\\\3x^2+8x-12<0\\\\D/4=4^2+3\cdot 12=52\; ,\; \; x_{1,2}=\frac{-4\pm \sqrt{52}}{3}=\frac{-4\pm 2\sqrt{13}}{3}" alt="\sqrt{x+2}-\sqrt{3x-1}>\sqrt{x-1}\\\\ODZ:\; \; \left \{ {{x\geq -2\; ,\; x\geq 1/3} \atop {x\geq 1}} \right. \; \; \to \; \; x\geq 1\\\\\underbrace {\sqrt{x+2}}_{\geq 0}>\underbrace {\sqrt{x-1}+\sqrt{3x-1}}_{\geq 0}\\\\x+2>(x-1)+2\cdot \sqrt{x-1}\cdot \sqrt{3x-1}+(3x-1)\\\\x+2>4x-2+2\cdot \sqrt{(x-1)(3x-1)}\\\\2\cdot \sqrt{(x-1)(3x-1)}<-3x+4\\\\4(x-1)(3x-1)<9x^2-24x+16\\\\4(3x^2-4x+1)<9x^2-24x+16\\\\3x^2+8x-12<0\\\\D/4=4^2+3\cdot 12=52\; ,\; \; x_{1,2}=\frac{-4\pm \sqrt{52}}{3}=\frac{-4\pm 2\sqrt{13}}{3}" align="absmiddle" class="latex-formula">
![x_1=\frac{-4-2\sqrt{13}}{3}\approx -3,74\; \; ;\; \; x_2=\frac{-4+2\sqrt{13}}{3}\approx 1,07\\\\(x-\frac{-4-2\sqrt{13}}{3})(x-\frac{-4+2\sqrt{13}}{3})<0\qquad \Big [\; (x+3,74)(x-1,07)<0\; \Big ]\\\\+++(-3,74)---(1,07)+++\qquad \Big [\; x\in (-3,74\; ;\; 1,07)\; \Big ]\\\\x\in \Big (\frac{-4-2\sqrt{13}}{3}\; ;\; \frac{-4+2\sqrt{13}}{3}\Big )\\\\\left \{ {{x\geq 1} \atop {x\in \left (\frac{-4-2\sqrt{13}}{3}}\; ;\; \frac{-4+2\sqrt{13}}{3}\right )} \right. \; \; \Rightarrow \; \; \; x\in \Big [\; 1\; ;\; \frac{-4+2\sqrt{13}}{3}\Big )](https://tex.z-dn.net/?f=x_1%3D%5Cfrac%7B-4-2%5Csqrt%7B13%7D%7D%7B3%7D%5Capprox%20-3%2C74%5C%3B%20%5C%3B%20%3B%5C%3B%20%5C%3B%20x_2%3D%5Cfrac%7B-4%2B2%5Csqrt%7B13%7D%7D%7B3%7D%5Capprox%201%2C07%5C%5C%5C%5C%28x-%5Cfrac%7B-4-2%5Csqrt%7B13%7D%7D%7B3%7D%29%28x-%5Cfrac%7B-4%2B2%5Csqrt%7B13%7D%7D%7B3%7D%29%3C0%5Cqquad%20%5CBig%20%5B%5C%3B%20%28x%2B3%2C74%29%28x-1%2C07%29%3C0%5C%3B%20%5CBig%20%5D%5C%5C%5C%5C%2B%2B%2B%28-3%2C74%29---%281%2C07%29%2B%2B%2B%5Cqquad%20%5CBig%20%5B%5C%3B%20x%5Cin%20%28-3%2C74%5C%3B%20%3B%5C%3B%201%2C07%29%5C%3B%20%5CBig%20%5D%5C%5C%5C%5Cx%5Cin%20%5CBig%20%28%5Cfrac%7B-4-2%5Csqrt%7B13%7D%7D%7B3%7D%5C%3B%20%3B%5C%3B%20%5Cfrac%7B-4%2B2%5Csqrt%7B13%7D%7D%7B3%7D%5CBig%20%29%5C%5C%5C%5C%5Cleft%20%5C%7B%20%7B%7Bx%5Cgeq%201%7D%20%5Catop%20%7Bx%5Cin%20%5Cleft%20%28%5Cfrac%7B-4-2%5Csqrt%7B13%7D%7D%7B3%7D%7D%5C%3B%20%3B%5C%3B%20%5Cfrac%7B-4%2B2%5Csqrt%7B13%7D%7D%7B3%7D%5Cright%20%29%7D%20%5Cright.%20%5C%3B%20%5C%3B%20%5CRightarrow%20%5C%3B%20%5C%3B%20%5C%3B%20x%5Cin%20%5CBig%20%5B%5C%3B%201%5C%3B%20%3B%5C%3B%20%5Cfrac%7B-4%2B2%5Csqrt%7B13%7D%7D%7B3%7D%5CBig%20%29 "x_1=\frac{-4-2\sqrt{13}}{3}\approx -3,74\; \; ;\; \; x_2=\frac{-4+2\sqrt{13}}{3}\approx 1,07\\\\(x-\frac{-4-2\sqrt{13}}{3})(x-\frac{-4+2\sqrt{13}}{3})<0\qquad \Big [\; (x+3,74)(x-1,07)<0\; \Big ]\\\\+++(-3,74)---(1,07)+++\qquad \Big [\; x\in (-3,74\; ;\; 1,07)\; \Big ]\\\\x\in \Big (\frac{-4-2\sqrt{13}}{3}\; ;\; \frac{-4+2\sqrt{13}}{3}\Big )\\\\\left \{ {{x\geq 1} \atop {x\in \left (\frac{-4-2\sqrt{13}}{3}}\; ;\; \frac{-4+2\sqrt{13}}{3}\right )} \right. \; \; \Rightarrow \; \; \; x\in \Big [\; 1\; ;\; \frac{-4+2\sqrt{13}}{3}\Big )")