Are the following systems of inequalities the same? Suppose $x, y \in \mathbb{R}$,  and $\mathcal{S_1}$ is a system of inequalities:
\begin{align*}
\mathcal{S_1} &= \begin{Bmatrix}
x  -  y  \geq 1\\
-x + 2y \geq 1\\
3x - 5y  \geq 2
\end{Bmatrix}\\
&= \begin{Bmatrix}
x  \geq 1 + y\\
x  \leq 2y-1\\
x  \geq \frac{2+5y}{3}
\end{Bmatrix}
\end{align*}
I eliminate $x$ from $S_1$ to obtain $S_2$:
\begin{align*}
\mathcal{S_2} &= \begin{Bmatrix}
1+y \leq 2y -1 \\
\frac{2+5y}{3} \leq 2y-1
\end{Bmatrix}\\
&= \begin{Bmatrix}
2 \leq y \\
2+5y \leq 6y-3
\end{Bmatrix}\\
&= \begin{Bmatrix}
2 \leq y \\
5 \leq y
\end{Bmatrix}
\end{align*}
Therefore, I know that $5 \leq y < \infty$. Let $S_3$ denote the following system of inequalities
\begin{align*}
\mathcal{S_3} &= \begin{Bmatrix}
x  -  y  \geq 1\\
-x + 2y \geq 1\\
3x - 5y  \geq 2\\
5 \leq y < \infty
\end{Bmatrix}
\end{align*}
My question is, is $S_3 = S_1$?

Yes, $S_3=S_1$. We need only to show that
$$5≤y<\infty$$
Therefore, we have,
$$3(-x+2y)+3x-5y≥3\times 1+2 $$
$$\iff y≥5.$$

But, this notation is problematic.
$$5≤y≤\infty$$
The correct one can be written as follows:
$$5≤y<\infty$$
Or,
$$y\in[5,+\infty)$$
