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(6)
![\dfrac{2x}{x+3}\geq 1\\ \\ \\
\dfrac{2x}{x+3}-1\geq 0\\ \\ \\
\dfrac{x-3}{x+3}\geq 0\\ \\ \\
(x-3)(x+3)\geq 0 \wedge x\neq -3\\ \\
x\in (-\infty,-3]\cup[3,+\infty)\wedge x\neq -3\\ \\
\boxed{x\in (-\infty,-3)\cup[3,+\infty)}](https://tex.z-dn.net/?f=%5Cdfrac%7B2x%7D%7Bx%2B3%7D%5Cgeq+1%5C%5C+%5C%5C+%5C%5C%0A%5Cdfrac%7B2x%7D%7Bx%2B3%7D-1%5Cgeq+0%5C%5C+%5C%5C+%5C%5C%0A%5Cdfrac%7Bx-3%7D%7Bx%2B3%7D%5Cgeq+0%5C%5C+%5C%5C+%5C%5C%0A%28x-3%29%28x%2B3%29%5Cgeq+0+%5Cwedge+x%5Cneq+-3%5C%5C+%5C%5C%0Ax%5Cin+%28-%5Cinfty%2C-3%5D%5Ccup%5B3%2C%2B%5Cinfty%29%5Cwedge+x%5Cneq+-3%5C%5C+%5C%5C%0A%5Cboxed%7Bx%5Cin+%28-%5Cinfty%2C-3%29%5Ccup%5B3%2C%2B%5Cinfty%29%7D%0A%0A%0A "\dfrac{2x}{x+3}\geq 1\\ \\ \\
\dfrac{2x}{x+3}-1\geq 0\\ \\ \\
\dfrac{x-3}{x+3}\geq 0\\ \\ \\
(x-3)(x+3)\geq 0 \wedge x\neq -3\\ \\
x\in (-\infty,-3]\cup[3,+\infty)\wedge x\neq -3\\ \\
\boxed{x\in (-\infty,-3)\cup[3,+\infty)}")
(7)
![\dfrac{x(x+5)}{x-3}\geq 0\\ \\
x(x+5)(x-3)\geq 0\wedge x\neq 3\\ \\
x\in [-5,0]\cup [3,+\infty)\wedge x\neq 3\\ \\
\boxed{x\in [-5,0]\cup (3,+\infty)}](https://tex.z-dn.net/?f=%5Cdfrac%7Bx%28x%2B5%29%7D%7Bx-3%7D%5Cgeq+0%5C%5C+%5C%5C+%0Ax%28x%2B5%29%28x-3%29%5Cgeq+0%5Cwedge+x%5Cneq+3%5C%5C+%5C%5C%0Ax%5Cin+%5B-5%2C0%5D%5Ccup+%5B3%2C%2B%5Cinfty%29%5Cwedge+x%5Cneq+3%5C%5C+%5C%5C%0A%5Cboxed%7Bx%5Cin+%5B-5%2C0%5D%5Ccup+%283%2C%2B%5Cinfty%29%7D "\dfrac{x(x+5)}{x-3}\geq 0\\ \\
x(x+5)(x-3)\geq 0\wedge x\neq 3\\ \\
x\in [-5,0]\cup [3,+\infty)\wedge x\neq 3\\ \\
\boxed{x\in [-5,0]\cup (3,+\infty)}")
(8)
![\dfrac{2}{x+1}\geq \dfrac{1}{x-2}\\ \\ \\
\dfrac{2}{x+1}- \dfrac{1}{x-2}\geq0\\ \\ \\
\dfrac{x-5}{(x+1)(x-2)}\geq 0\\ \\ \\
(x-5)(x+1)(x-2)\geq 0\wedge x\neq -1\wedge x\neq 2\\ \\ \\
x\in [-1,2]\cup[5,+\infty)\wedge x\neq -1\wedge x\neq 2\\ \\ \\
\boxed{x\in (-1,2)\cup[5,+\infty)}](https://tex.z-dn.net/?f=%5Cdfrac%7B2%7D%7Bx%2B1%7D%5Cgeq+%5Cdfrac%7B1%7D%7Bx-2%7D%5C%5C+%5C%5C+%5C%5C%0A%5Cdfrac%7B2%7D%7Bx%2B1%7D-+%5Cdfrac%7B1%7D%7Bx-2%7D%5Cgeq0%5C%5C+%5C%5C+%5C%5C%0A%5Cdfrac%7Bx-5%7D%7B%28x%2B1%29%28x-2%29%7D%5Cgeq+0%5C%5C+%5C%5C++%5C%5C%0A%28x-5%29%28x%2B1%29%28x-2%29%5Cgeq+0%5Cwedge+x%5Cneq+-1%5Cwedge+x%5Cneq+2%5C%5C+%5C%5C+%5C%5C%0Ax%5Cin+%5B-1%2C2%5D%5Ccup%5B5%2C%2B%5Cinfty%29%5Cwedge+x%5Cneq+-1%5Cwedge+x%5Cneq+2%5C%5C+%5C%5C+%5C%5C%0A%5Cboxed%7Bx%5Cin+%28-1%2C2%29%5Ccup%5B5%2C%2B%5Cinfty%29%7D%0A "\dfrac{2}{x+1}\geq \dfrac{1}{x-2}\\ \\ \\
\dfrac{2}{x+1}- \dfrac{1}{x-2}\geq0\\ \\ \\
\dfrac{x-5}{(x+1)(x-2)}\geq 0\\ \\ \\
(x-5)(x+1)(x-2)\geq 0\wedge x\neq -1\wedge x\neq 2\\ \\ \\
x\in [-1,2]\cup[5,+\infty)\wedge x\neq -1\wedge x\neq 2\\ \\ \\
\boxed{x\in (-1,2)\cup[5,+\infty)}")
(7)
(8)
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