Consider the inequality \( \frac{x}{-2} \leq-2.5 \). Part A: Would the inequality symbol reverse when the inequality is solved? Explain why or why not. Part B: Solve the inequality, and state the solution set. Part C: Would the process of solving the inequality be different if the inequality was \( -2 x \leq-2.5 \) ? If so, explain how. Then solve the inequality, and state the solution set.

When solving inequalities, the inequality symbol reverses if you multiply or divide both sides by a negative number. In the case of \( \frac{x}{-2} \leq -2.5 \), since we're dividing by -2 (a negative), we indeed reverse the inequality sign. To solve \( \frac{x}{-2} \leq -2.5 \), we first multiply both sides by -2 (remembering to reverse the inequality sign): \( x \geq 5 \). Therefore, the solution set is \( x \in [5, \infty) \). For the inequality \( -2x \leq -2.5 \), the process differs slightly because we are also multiplying by -1 to isolate \( x \). In this case, we indeed reverse the inequality sign. To solve, we can divide by -2: \( x \geq 1.25 \). Thus, the solution set is \( x \in [1.25, \infty) \). Historical Background: Inequalities have been used for centuries in mathematics, dating back to ancient civilizations where they were essential for solving practical problems in commerce and engineering. The formal notation we use today evolved during the late 19th century, with mathematicians like André-Marie Ampère playing a key role in refining these concepts. Real-World Application: Understanding inequalities is crucial in various real-life situations, such as finance, where you might need to determine how much money you need to save or invest to achieve a target financial goal. For example, if you're planning a budget and want to limit expenses to a certain amount, inequalities help set clear boundaries.