Linear Inequalities in Real-Life Scenarios?

📌 CONCEPT: A linear inequality is a mathematical statement that compares two expressions, often representing a constraint or a limitation in real-life scenarios. In this case, the inequality 3x + 2 < 18 represents the daily operating hours of a library. The task is to design a new library schedule while adhering to this inequality and a maximum operating hour limit.

📐 RULE / FORMULA: To find the feasible operating hours for the new library, we need to solve the inequality 3x + 2 < 18 for x. This will give us the range of operating hours that satisfy the given inequality.

💡 WORKED EXAMPLE: Let's find the feasible operating hours for the new library. First, we solve the inequality 3x + 2 < 18 for x: 3x < 16, x < −2₄₃₃₄₁. Since x represents the number of hours, we discard the negative value and consider x < ₅₃₃₃₂. Now, we need to ensure that the maximum operating hours are not exceeded. Given that the maximum operating hours allowed are 12 hours, we can propose a suitable operating schedule of 11 hours or less to adhere to the given inequality.

⚠️ COMMON MISTAKE: Students often forget to consider the context of the problem, leading to incorrect solutions. In this case, they may ignore the maximum operating hour limit, resulting in an unsuitable library schedule.