Szczegółowe wyjaśnienie:
[tex]1.\ |x-2|=3\iff x-2=3\ \vee\ x-2=-3\qquad|+2\\\\\huge\boxed{x=5\ \vee\ x=-1}[/tex]
[tex]2.\ |3-x|=4\iff3-x=4\ \vee\ 3-x=-4\qquad|-3\\\\-x=1\ \vee\ -x=-7\qquad|\cdot(-1)\\\\\huge\boxed{x=-1\ \vee\ x=7}[/tex]
[tex]3.\ |2x+4|=6\iff2x+4=6\ \vee\ 2x+4=-6\qquad|-4\\\\2x=2\ \vee\ 2x=-10\qquad|:2\\\\\huge\boxed{x=1\ \vee\ x=-5}[/tex]
[tex]4.\ |x-1|+|1-x|=4\\\\|x-1|+|-1(x-1)|=4\\\\|x-1|+|-1|\cdot|x-1|=4\\\\|x-1|+|x-1|=4\\\\2|x-1|=4\qquad|:2\\\\|x-1|=2\iff x-1=2\ \vee\ x-1=-2\qquad|+1\\\\\huge\boxed{x=3\ \vee\ x=-1}[/tex]
[tex]5.\ \sqrt{x^2+6x+9}=5\\\\\sqrt{x^2+2\cdot x\cdot3+3^2}=5\qquad|(a+b)^2=a^2+2ab+b^2\\\\\sqrt{(x+3)^2}=5\iff|x+3|=5\iff x+3=5\ \vee\ x+3=-5\qquad|-3\\\\\huge\boxed{x=2\ \vee\ x=-8}[/tex]
[tex]6.\ |2x+8|+\sqrt{x^2+8x+16}=1\\\\|2(x+4)|+\sqrt{x^2+2\cdot x\cdot 4+4^2}=1\qquad|(a+b)^2=a^2+2ab+b^2\\\\2|x+4|+\sqrt{(x+4)^2}=1\iff2|x+4|+|x+4|=1\\\\3|x+4|=1\qquad|:3\\\\|x+4|=\dfrac{1}{3}\iff x+4=\dfrac{1}{3}\ \vee\ x+4=-\dfrac{1}{3}\qquad|-4\\\\\huge\boxed{x=-3\dfrac{2}{3}\ \vee\ x=-4\dfrac{1}{3}}[/tex]
