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1-(x+3)$\ge$4-2x

$\small{ \begin{array}{lrcll} & 1-(x+3) &\ge& 4-2x \\ \\ \hline \\ (1) & 1-(x+3) & = & 4-2x \\ & 1 - x - 3 &=& 4 - 2x \\ & -2 - x &=& 4 - 2x \qquad & | \qquad + 2x\\ & -2 + x &=& 4 \qquad & | \qquad +2\\ & \mathbf{x} & \mathbf{=} & \mathbf{6 }\\ \\ \hline \\ (2) & 1-(x+3) & > & 4-2x \\ & 1 - x - 3 & > & 4 - 2x \\ & -2 - x & > & 4 - 2x \qquad & | \qquad + 2x\\ & -2 + x &> & 4 \qquad & | \qquad +2\\ & \mathbf{x} & \mathbf{>} & \mathbf{6 }\\ \\ \hline \\ (1) + ( 2) & x = 6 ~ \text{ and } ~ x > 6 \\ & \mathbf{ x \ge 6 } \end{array} }$