Solving Inequalities: Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of inequalities, specifically tackling the problem presented by Silva Villamar. We'll break down how to find the solution set for a given inequality and, most importantly, how to express that solution in interval notation. Trust me, once you get the hang of it, it's like unlocking a secret mathematical code! So, let's jump right in and make inequalities our new best friends.
Understanding Inequalities and Solution Sets
Before we even look at the specific problem, let's make sure we're all on the same page about what inequalities actually are. Unlike equations, which state that two expressions are equal, inequalities show a relationship between two expressions where they might not be equal. We use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to). The goal when solving an inequality is the same as solving an equation: to isolate the variable. However, there's one very important rule to remember: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is crucial! Forget this, and your solution set will be completely off. Now that we've refreshed our understanding of inequalities, let's talk about solution sets. The solution set of an inequality is simply the set of all values that make the inequality true. This is where things get interesting because, unlike equations that often have a single solution (or a few), inequalities usually have an infinite number of solutions. Think about it: if x > 5, then 6, 7, 8, 100, 1000, and an infinite number of other values all satisfy the inequality. This is why we need a way to represent this infinite set of solutions, and that's where interval notation comes in handy. To properly solve an inequality, we must carefully isolate the variable. This often involves using the distributive property to eliminate parentheses and combining like terms. We then use inverse operations to isolate the variable. Remember the golden rule: If you multiply or divide by a negative number, flip the inequality sign! After isolating the variable, you will have the solution set in inequality notation (e.g., x < 3). The final step is to convert this into interval notation. This gives us a clear and concise way to represent the infinite set of solutions.
Expressing Solutions in Interval Notation
Okay, so we've solved our inequality and have a solution set. Now comes the art of expressing that solution in interval notation. This might seem a bit strange at first, but it's actually a very efficient way to represent a range of numbers. Interval notation uses parentheses and brackets to indicate whether the endpoints of an interval are included in the solution or not. A parenthesis '(' or ')' means that the endpoint is not included (it's an open interval), while a bracket '[' or ']' means that the endpoint is included (it's a closed interval). For example, if our solution is x > 5, we would write this in interval notation as (5, ∞). Notice the parenthesis next to the 5, indicating that 5 is not included in the solution (since x is strictly greater than 5). The infinity symbol ∞ always gets a parenthesis because infinity isn't a specific number, so we can't actually
