What is the largest $x$ to satisfy these three inequalities?

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The numbers \(x\) and \(y\) satisfy the following inequalities \[\begin{align*} 2x+3y &\le 23, \\ x+2 &\le 3y, \\ 3y+1 &\le 4x. \end{align*}\]

The largest possible value of \(x\) is

1. \(6\),

2. \(7\),

3. \(8\),

4. \(9\).

Can we draw a diagram? Where are all the inequalities satisfied?

Or would working purely algebraically be more useful?

Where in this region does the largest value of \(x\) occur?