Question

Решить неравенство $sqrt{x+8} textless x+2$

Answer 1

$sqrt{x+8} textless x+2$

$left { {{x+8 geq 0} atop {x+2 textgreater 0}} atop {( sqrt{x+8})^2 textless (x+2)^2 } right.$

$left { {{x geq -8} atop {x textgreater -2}} atop {x+8 textless x^2+4x+4 } right.$

$left { {{x geq -8} atop {x textgreater -2}} atop {-x^2-3x+4 textless 0 } right.$

$left { {{x geq -8} atop {x textgreater -2}} atop {x^2+3x-4 textgreater 0 } right.$

$x^2+3x-4=0$

$D=3^2-4*1*(-4)=25$

$x_1= frac{-3+5}{2} =1$

$x_2= frac{-3-5}{2} =-4$

$left { {{x geq -8} atop {x textgreater -2}} atop {(x-1)(x+4) textgreater 0 } right.$

----------[-8]-------------------------------------------
              /////////////////////////////////////////////
-----------------------(-2)------------------------------
                            ///////////////////////////////
         +                        -                     +
----------------(-4)------------------(1)---------------
//////////////////                         //////////////////

Ответ: $(1;+$ ∞ $)$