Are the following systems of inequalities the same?

Question

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 $S_3 = S_1$?

Answer 1

Yes, $S_3=S_1$. We need only to show that

$$5≤y<\infty$$

Therefore, we have,

$$3(-x+2y)+3x-5y≥3\times 1+2 $$

$$\iff y≥5.$$

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But, this notation is problematic.

$$5≤y≤\infty$$

The correct one can be written as follows:

$$5≤y<\infty$$

Or,

$$y\in[5,+\infty)$$