1. **State the problem:** We have two types of tablets, Vita A and Vita B. - Vita A contains 30 mg iron and 20 mg vitamin C per tablet. - Vita B contains 10 mg iron and 30 mg vitamin C per tablet. The daily needs are: - At most 180 mg iron (iron intake $\leq 180$) - At least 1500 mg vitamin C (vitamin C intake $\geq 1500$) We want to find the system of inequalities representing these constraints. 2. **Define variables:** Let $x$ = number of Vita A tablets Let $y$ = number of Vita B tablets 3. **Write inequalities for iron and vitamin C:** - Iron intake: $30x + 10y \leq 180$ (since at most 180 mg iron) - Vitamin C intake: $20x + 30y \geq 1500$ (since at least 1500 mg vitamin C) 4. **Non-negativity constraints:** $x \geq 0, y \geq 0$ because number of tablets cannot be negative. 5. **Check options:** - Option a: $30x + 10y \leq 1800; 20x + 30y \geq 1500; x,y \geq 0$ - Iron inequality has 1800 instead of 180, so incorrect. - Option b: $30x + 10y \geq 1800; 20x + 30y \geq 1500$ - Iron inequality reversed and wrong number, incorrect. - Option c: $10x + 30y \leq 1800; 30x + 20y \geq 1500$ - Coefficients swapped, incorrect. - Option d: $30x + 10y \leq 1500; 20x + 30y \leq 1800$ - Vitamin C inequality reversed and numbers swapped, incorrect. - Option e: None of the given options is correct. 6. **Conclusion:** None of the options matches the correct system: $$ \begin{cases} 30x + 10y \leq 180 \\ 20x + 30y \geq 1500 \\ x \geq 0, y \geq 0 \end{cases} $$ Therefore, the correct answer is option e. **Final answer:** e. None of the given options is correct.