Сколько корней имеет уравнение |||x?1|+2|?3|=x+4?

8/Задание

№ 3:

РЕШЕНИЕ:

|||x-1|+2|-3|=x+4

ОДЗ: $x \geq -4$

$\left[\begin{array}{l}&amp;#10;||x-1|+2|-3=x+4 \\||x-1|+2|-3=-x-4 \end{array},x \geq -4$
$\left[\begin{array}{l} ||x-1|+2|=x+7 \\||x-1|+2|=-x-1 \end{array},x \geq&amp;#10;-4$
$\left[\begin{array}{l} \left[\begin{array}{l} |x-1|+2=x+7 \\ |x-1|+2=-x-7&amp;#10;\end{array},x \geq -7 \\ \left[\begin{array}{l} |x-1|+2=-x-1 \\ |x-1|+2=x+1&amp;#10;\end{array} , x \leq -1 \end{array},x \geq -4$
$\left[\begin{array}{l} \left[\begin{array}{l} |x-1|=x+5 \\ |x-1|=-x-9&amp;#10;\end{array},x \geq -7 \\ \left[\begin{array}{l} |x-1|=-x-3 \\ |x-1|=x-1&amp;#10;\end{array} , x \leq -1 \end{array},x \geq -4$
|x-1|=x-1 по свойству модуля х>=1
$\left[\begin{array}{l} \left[\begin{array}{l} \left[\begin{array}{l}&amp;#10;x-1=x+5 \\ x-1=-x-5 \end{array},x \geq -5 \\ \left[\begin{array}{l} x-1=-x-9 \\&amp;#10;x-1=x+9 \end{array},x \leq -9 \end{array},x \geq -7 \\ \left[\begin{array}{l}&amp;#10;\left[\begin{array}{l} x-1=-x-3 \\ x-1=x+3 \end{array},x \leq -3 \\ x \geq 1&amp;#10;\end{array} , x \leq -1 \end{array},x \geq -4$
x<=-9 и x>=-7 несовместные условия
x>=-5 поглощает условие x>=-7
$\left[\begin{array}{l} \left[\begin{array}{l} \left[\begin{array}{l}&amp;#10;x-1=x+5 \\ x-1=-x-5 \end{array},x \geq -5 \end{array} \\ \left[\begin{array}{l}&amp;#10;\left[\begin{array}{l} x-1=-x-3 \\ x-1=x+3 \end{array},x \leq -3 \\ x \geq 1&amp;#10;\end{array} , x \leq -1 \end{array},x \geq -4$
x-1=x+5 и x-1=x+3 не дают решений
$\left[\begin{array}{l} x-1=-x-5,x \geq -5 \\ \left[\begin{array}{l}&amp;#10;x-1=-x-3,x \leq -3&nbsp; \\ x \geq 1 \end{array} , x \leq -1 \end{array},x \geq&amp;#10;-4$
$\left[\begin{array}{l} 2x=-4,x \geq -5 \\ \left[\begin{array}{l} 2x=-2,x&amp;#10;\leq -3 \\ x \geq 1 \end{array} , x \leq -1 \end{array},x \geq -4$
$\left[\begin{array}{l} x=-2,x \geq -5 \\ \left[\begin{array}{l} x=-1,x&amp;#10;\leq -3 \\ x \geq 1 \end{array} , x \leq -1 \end{array},x \geq -4$
x=-1 и x<=-3 несовместны
$\left[\begin{array}{l} x=-2,x \geq -5 \\ x \geq 1 , x \leq -1&amp;#10;\end{array},x \geq -4$
x>=1 и x<=-1 несовместны
$x=-2,x \geq -5,x \geq -4$
x=-2 удовлетворяет условиям