Question

$sqrt{x-5}+ sqrt{10-x} textless 3$ 

Answer 1

$sqrt{x-5}+ sqrt{10-x} textless 3$
ОДЗ:
$left { {{x-5 geq 0} atop {10-x geq 0}} right.$
$left { {{x geq 5} atop {x leq 10}} right.$
$x$ ∈ $[5;10]$

$(sqrt{x-5}+ sqrt{10-x})^2 textless 3^2$
${x-5}+10-x}+2 sqrt{(x-5)(10-x)} textless 9$
$5+2 sqrt{(x-5)(10-x)} textless 9$
$2 sqrt{(x-5)(10-x)} textless 4$
$sqrt{(x-5)(10-x)} textless 2$
$( sqrt{(x-5)(10-x)})^2 textless 2^2$
$(x-5)(10-x)} textless 4$
$10x- x^{2} -50+5x-4} textless 0$
$- x^{2}+15x-54} textless 0$
$x^{2}-15x+54} textgreater 0$
$D=(-15)^2-4*1*54=9$
$x_1= frac{15+3}{2}=9$
$x_2= frac{15-3}{2}=6$

-----+-----(6)------ - ----------(9)------+-------
//////////////                               ///////////////
------[5]-------------------------------[10]--------
           ////////////////////////////////////

Ответ: [5;6) ∪ $(9;10]$