= Solution
If $x_1,x_2$ are integral, the derived surplus and slacks
$$
x_3=2x_1+x_2-6,\qquad x_4=x_2-2x_1,\qquad x_5=8-x_1-2x_2
$$
are also nonnegative integers. Therefore an all-integer <Gomory fractional cut> can be taken from the final tableau's slack row
$$
x_4+\frac53x_3+\frac43x_5=\frac23.
$$
For a nonnegative integer dictionary row $x_B+\sum_j a_jx_j=b$, the quantity $x_B+\sum_j\lfloor a_j\rfloor x_j$ is an integer at most $b$, so it is at most $\lfloor b\rfloor$. Subtracting from the row proves the cut $\sum_j\{a_j\}x_j\geq\{b\}$. Applying it here gives
$$
\boxed{2x_3+x_5\geq2.}
$$
But the original row and $x_4\geq0$ give $5x_3+4x_5\leq2$. Nonnegativity then gives
$$
2x_3+x_5\leq\frac25(5x_3+4x_5)\leq\frac45<2.
$$
This contradicts the valid cut. Therefore \b[the integer program is infeasible], certified by one <Gomory fractional cut>. It is essential to use the PDF's constraint: the altered TeX problem would have feasible integer points.
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