$P_0$ (Root problem) $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(\tfrac{15}4,\tfrac94)$ with $z=41.25$. Both variables $x_1$ and $x_2$ are fractional. Branch on $x_1$, splitting into $x_1\leq\lfloor\tfrac{15}4\rfloor=3$ and $x_1\geq\lceil\tfrac{15}4\rceil=4$.

$B_{up}=41.25$ and $B_{low}=-\infty$

$P_{1.2}$ with $x_1\ge 4$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 \ge 4\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(4,\tfrac95)$ with $z=41$. The variable $x_2$ is fractional. Branch on $x_2$, splitting into $x_2\leq\lfloor\tfrac95\rfloor=1$ and $x_2\geq\lceil\tfrac95\rceil=2$.

$B_{up}=41.25$ and $B_{low}=39$

$P_{1.1}$ with $x_1\le 3$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 \le 3\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(3,3)$ with $z=39$. The optimum is integer feasible, so $B_{low}=39$. This branch is fully resolved.

$B_{up}=41.25$ and $B_{low}=39$

$P_{2.1}$ with $x_1\ge 4, x_2\le 1$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 \ge 4\ & x_2 \le 1\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(\tfrac{40}9,1)$ with $z=40.56$. The variable $x_1$ is fractional. Branch on $x_1$, splitting into $x_1\leq\lfloor\tfrac{40}9\rfloor=4$ and $x_1\geq\lceil\tfrac{40}9\rceil=5$.

$B_{up}=41.25$ and $B_{low}=39$

$P_{2.2}$ with $x_1\ge 4, x_2\ge 2$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 \ge 4\ & x_2 \ge 2\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation is infeasible. The constraints $x_1 \geq 4$ and $x_2 \geq 2$ together require $9x_1 + 5x_2 \geq 9(4) + 5(2) = 46 > 45$.

$P_{3.1}$ with $x_1\ge 4, x_2\le 1, x_1\le 4$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 = 4\ & x_2 \le 1\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(4,1)$ with $z=37$. The optimum is integer feasible, but $37$ does not improve the incumbent $B_{low}=39$.

$B_{up}=41.25$ and $B_{low}=39$

$P_{3.2}$ with $x_1\ge 4, x_2\le 1, x_1\ge 5$ $$ \begin{alignat*}{3} \max_{\boldsymbol{x}}\quad& 8x_1+5x_2 \ \text{s.t.}\quad & x_1 + x_2 \le 6 \ & 9x_1+5x_2 \le 45 \ & x_1 \ge 5\ & x_2 \le 1\ & x_1,x_2 \ge 0, ; x_1,x_2 \in \mathbb{Z} \end{alignat*} $$ The LP relaxation has the optimum solution $\boldsymbol{x}^*=(5,0)$ with $z=40$. The optimum is integer feasible, and improve the incumbent $B_{low}=39$.

$B_{up}=41.25$ and $B_{low}=40$