Question

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Answer 1

$( frac{10}{5x-21} + frac{5x-21}{10} )^2 leq frac{25}{4} ; ;quad ODZ:quad xne frac{21}{5}\\t=frac{10}{5x-21}; ,; ; (t+frac{1}{t})^2 leq frac{25}{4}; ,\\(frac{t^2+1}{t})^2-(frac{5}{2})^2 leq 0\\(frac{t^2+1}{t}-frac{5}{2})(frac{t^2+1}{t}+frac{5}{2}) leq 0\\ frac{2t^2-5t+2}{2t} cdot frac{2t^2+5t+2}{2t} leq 0\\ frac{2(t-2)(t-frac{1}{2})cdot 2(t+2)(t+frac{1}{2})}{4t^2} leq 0quad quad frac{(t-2)(t-frac{1}{2})(t+2)(t+frac{1}{2})}{t^2} leq 0$

$+++[-2]---[-frac{1}{2}]+++(0)+++[frac{1}{2}]---[2]+++\\t leq -2; ;; -frac{1}{2} leq t textless 0; ;; 0 textless t leq frac{1}{2}; ;; t geq 2\\ 1)frac{10}{5x-21} leq -2; ;; frac{10x-32}{5x-21} leq 0; ;; quad xin [3,2; ;; 4,2)\\ 2)left { {{ frac{10}{5x-21} geq - frac{1}{2} } atop { frac{10}{5x-21} textless 0}} right. ; left { {{ frac{5-1x}{2(5x-21)} geq 0} atop {5x-21 textless 0}} right. ; left { {{xin (-infty ,;, 0,2]cup (4,2, ;+infty )} atop {xin (-infty ;; 4,2)}} right.$$xin (-infty ;; 0,2]\\3); left { {{ frac{10}{5x-21} textgreater 0 } atop { frac{10}{5x-21} leq frac{1}{2} }} right. ; left { {{5x-21 textgreater 0} atop {frac{1-5x}{2(5x-21)} leq 0}} right. ; left { {{x textgreater 4,2} atop {xin (-infty ;; 0,2]cup (4,2; ;+infty )}} right. ; to ; xin (4,2; ;; +infty )\\4); frac{10}{5x-21} geq 2; ;; frac{52-10x}{5x-21} geq 0; ;; frac{10x-52}{5x-21} leq 0; ;; xin (4,2; ;; 5,2]\\5); xin (-infty ;; 0,2]cup [3,2; ;; 4,2)cup (4,2; ;5,2]$