1. **State the problem:** We are given two inequalities involving variables $\beta_{ji}$, $\theta_{ji}$, $x_{ij}$, and constants $b_{ij}$, $A_{ij}$: $$\beta_{ji} x_{ij} \leq \theta_{ji} b_{ij}$$ and for $i \neq j$: $$\beta_{ji} A_{ij} \leq \theta_{ji} b_{ij}$$ Our goal is to understand these inequalities and possibly express one variable in terms of others or analyze the conditions. 2. **Analyze the first inequality:** $$\beta_{ji} x_{ij} \leq \theta_{ji} b_{ij}$$ Assuming $x_{ij} \neq 0$, divide both sides by $x_{ij}$ (noting the direction of the inequality depends on the sign of $x_{ij}$): - If $x_{ij} > 0$: $$\beta_{ji} \leq \frac{\theta_{ji} b_{ij}}{x_{ij}}$$ - If $x_{ij} < 0$: $$\beta_{ji} \geq \frac{\theta_{ji} b_{ij}}{x_{ij}}$$ - If $x_{ij} = 0$: The inequality reduces to $0 \leq \theta_{ji} b_{ij}$. 3. **Analyze the second inequality for $i \neq j$:** $$\beta_{ji} A_{ij} \leq \theta_{ji} b_{ij}$$ Similarly, assuming $A_{ij} \neq 0$: - If $A_{ij} > 0$: $$\beta_{ji} \leq \frac{\theta_{ji} b_{ij}}{A_{ij}}$$ - If $A_{ij} < 0$: $$\beta_{ji} \geq \frac{\theta_{ji} b_{ij}}{A_{ij}}$$ - If $A_{ij} = 0$: The inequality reduces to $0 \leq \theta_{ji} b_{ij}$. 4. **Summary:** These inequalities restrict $\beta_{ji}$ based on $\theta_{ji}$, $b_{ij}$ and the parameters $x_{ij}$, $A_{ij}$. The signs of $x_{ij}$ and $A_{ij}$ dictate inequality directions when dividing. No further simplification is possible without more information about the variables or their ranges.