Question

Подробное решение неравенств

Answer 1

$1)quad frac{x+3}{sqrt{x+1}} textgreater 3; ,; ; ODZ:; ; x+1 textgreater 0; ,; ; x textgreater -1\\frac{x+3-3sqrt{x+1}}{sqrt{x+1}} textgreater 0\\Tak; kak; sqrt{x+1} textgreater 0,; to; ; ; ; x+3-3sqrt{x+1} textgreater 0; ,\\3sqrt{x+1} textless x+3; ; ; Rightarrow ; ; ; left { {{x+1 textgreater 0; ,; x+3 textgreater 0} atop {9(x+1) textless (x+3)^2}} right. \\ left { {{x textgreater -1} atop {9x+9 textless x^2+6x+9}} right. ; left { {{x textgreater -1} atop {x^2-3x textgreater 0}} right. ; left { {{x textgreater -1} atop {x(x-3) textgreater 0}} right. ; left { {{x textgreater -1} atop {x textless 0; ili; x textgreater 3}} right.$

$xin (-1,0)cup (3,+infty )$

$2)quad frac{x+4}{sqrt{x+1}} textless 4; ,; ; ODZ:; ; x textgreater -1\\frac{x+4-4sqrt{x+1}}{sqrt{x+1}} textless 0\\sqrt{x+1} textgreater 0; ; to ; ; ; x+4-4sqrt{x+1} textless 0\\4sqrt{x+1} textgreater x+4; ; ; Leftrightarrow ; ; left { {{x+4 textless 0} atop {x+1 textgreater 0}} right. ; ili; ; left { {{x+4 geq 0} atop {16(x+1) textgreater (x+4)^2}} right. \\ left { {{x textless -4} atop {x textgreater -1}} right. ; ; ili; ; left { {{x geq -4} atop {16x+16 textgreater x^2+8x+16}} right.$

$xin varnothing ; ; ; ili; ; ; left { {{x geq -4} atop {x^2-8x<0}} right.$

$qquad quad left { {{x geq -4} atop {x(x-8) textless 0}} right. ; left { {{x geq -4} atop {0 textless x textless 8}} right. ; ; to ; ; xin (0,8); ; -; otvet$

$3)quad frac{x-9}{sqrt{x+9}} textless 1; ,; ; ; ODZ:; x textgreater -9 \\ frac{x-9-sqrt{x+9}}{sqrt{x+9}} textless 0; ; to ; ; x-9-sqrt{x+9} textless 0\\sqrt{x+9} textgreater x-9; ; Leftrightarrow ; ; left { {{x-9 textless 0} atop {x+9 textgreater 0}} right. ; ; ili; ; left { {{x-9 geq 0} atop {x+9 textgreater (x-9)^2}} right. \\ left { {{x textless 9} atop {x textgreater -9}} right. ; ; ili; ; left { {{x geq 9} atop {x+9 textgreater x^2-18x+81}} right. \\-9 textless x textless 9; ; ili; ; left { {{x geq 9} atop {x^2-19x+72 textless 0}} right. ; to$

$left { {{x geq 9} atop {frac{1}{2}(19-sqrt{73}) textless x textless frac{1}{2}(19+sqrt{73})}} right. ; ; to ; ; xin [, 9,; frac{1}{2}(19+sqrt{73}); )\\Otvet:; ; xin (-9,; frac{1}{2}(19+sqrt{73}); )$

$4)quad frac{x-6}{sqrt{x+9}} textgreater 2; ,; ; ; ODZ:; x textgreater -9\\ frac{x-6-2sqrt{x+9}}{sqrt{x+9}} textgreater 0; ; to ; ; x-6-2sqrt{x+9} textgreater 0\\2sqrt{x+9} textless x-6; ; Leftrightarrow ; ; left { {{x+9 textgreater 0; ,; x-6 textgreater 0} atop {4(x+9) textless (x-6)^2}} right. \\ left { {{x textgreater -9; ,; x textgreater 6} atop {4x+36 textless x^2-12x+36}} right. ; left { {{x textgreater 6} atop {x^2-16x textgreater 0}} right. ; left { {{x textgreater 6} atop {x(x-16) textgreater 0}} right. ; left { {{x textgreater 6} atop {x textless 0; ili; x textgreater 16}} right. ; Rightarrow \\xin (16,+infty ); -; otvet$