\(\left|-4x\right|=2\left(x+1\right)\)

\(-4x< 0\Leftrightarrow x>0\)

\(-4x\ge0\Leftrightarrow x\le0\)

Voi: \(x>0\)

\(pt\Leftrightarrow4x=2\left(x+1\right)\Leftrightarrow x=1\left(tm\right)\)

Voi : \(x\le0\)

\(pt\Leftrightarrow-4x=2\left(x+1\right)\Leftrightarrow x=-\frac{1}{3}\left(tm\right)\)

Vay phuong trinh nay co tap nghiem la: \(S=\left\{1;-\frac{1}{3}\right\}\)

Khong chac :<

Ta có: P =  AB = \(\frac{3}{4}\cdot\left(\frac{x+3}{x-3}\right)=\frac{3\left(x+3\right)}{4\left(x-3\right)}=\frac{3x+9}{4x-12}\)

=> 4P = \(\frac{4\left(3x+9\right)}{4x-12}=\frac{4\left(3x+9\right)}{4\left(x-3\right)}=\frac{3x+9}{x-3}=\frac{3\left(x-3\right)+18}{x-3}=3+\frac{18}{x-3}\)

Để P \(\in\)Z <=> 4P \(\in\)Z <=> 18 \(⋮\)x - 3

<=> x - 3 \(\in\)Ư(18) = {1; -1; 2; -2; 3; -3; 6; -6; 9; -9; 18; -18}

Lập bảng:

vậy ....

Thay \(2016=xyz\)vào biểu thức ta được

\(A=\frac{x^2yz}{xy+x^2yz+xyz}+\frac{y}{yz+y+xyz}+\frac{z}{xz+z+1}\)

   \(=\frac{x^2yz}{xy\left(1+xz+z\right)}+\frac{y}{y\left(z+1+xz\right)}+\frac{z}{xz+z+1}\)

   \(=\frac{xz}{xz+z+1}+\frac{1}{xz+z+1}+\frac{z}{xz+z+1}=\frac{xz+z+1}{xz+z+1}=1\)

Vậy \(A=1\)

Vì \(xyz=2016\)

\(\Rightarrow A=\frac{2016x}{xy+2016x+2016}+\frac{y}{yz+y+2016}+\frac{z}{xz+z+1}\)

\(=\frac{xyz.x}{xy+xyz.x+xyz}+\frac{y}{yz+y+xyz}+\frac{z}{xz+z+1}\)

\(=\frac{x^2yz}{xy+x^2yz+xyz}+\frac{y}{y\left(z+1+xz\right)}+\frac{z}{xz+z+1}\)

\(=\frac{x^2yz}{xy\left(1+xz+z\right)}+\frac{1}{xz+z+1}+\frac{z}{xz+z+1}\)

\(=\frac{xz}{xz+z+1}+\frac{1}{xz+z+1}+\frac{z}{xz+z+1}\)

\(=\frac{xz+1+z}{xz+z+1}=1\)

Đặt \(A=\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)

\(\Rightarrow A=\left(x+2\right)\left(x+5\right)\left(x+3\right)\left(x+4\right)-24\)

         \(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)

Đặt \(x^2+7x+11=t\)

\(\Rightarrow A=\left(t-1\right)\left(t+1\right)-24=t^2-1-24=t^2-25=\left(t-5\right)\left(t+5\right)\)

        \(=\left(x^2+7x+11-5\right)\left(x^2+7x+11+5\right)=\left(x^2+7x+6\right)\left(x^2+7x+16\right)\)

        \(=\left(x+1\right)\left(x+6\right)\left(x^2+7x+16\right)\)