Solving Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of inequalities and tackling a specific problem that might seem a bit daunting at first. But trust me, once you grasp the fundamental concepts, you'll be solving these like a pro. We're going to break down the inequality $-10x + 2 \geq -11x + 13$ and figure out the solution set, expressing it in that fancy interval notation. So, buckle up and let's get started!

Understanding Inequalities: More Than Just Equations

Before we jump into solving, let's quickly recap what inequalities are all about. Unlike equations, which are all about finding exact values that make both sides equal, inequalities deal with relationships where one side is greater than, less than, greater than or equal to, or less than or equal to the other side. Think of it like a balancing scale, but instead of perfectly balanced, one side is heavier or lighter. In our specific inequality, $-10x + 2 \geq -11x + 13$, the "$\geq$" symbol tells us that the expression on the left side ($-10x + 2$) is greater than or equal to the expression on the right side ($-11x + 13$). This "greater than or equal to" symbol is crucial, and we'll see how it impacts our final solution.

Why are inequalities important, you ask? Well, they pop up everywhere in real-world scenarios! Imagine you're budgeting your expenses – you want your spending to be less than or equal to your income. Or perhaps you're calculating the minimum speed you need to travel to reach your destination on time. Inequalities are the mathematical tools that help us describe these kinds of constraints and limitations. They allow us to work with ranges of values rather than just single, precise answers. This is why understanding inequalities is super useful not just in math class, but also in everyday life.

Now, let's talk about the solution set. The solution set of an inequality is the collection of all values that make the inequality true. In other words, it's the range of numbers that, when plugged into the variable (in our case, x), will satisfy the given relationship. This is where interval notation comes in handy. Interval notation is a concise way to represent a set of numbers using brackets and parentheses. We'll delve deeper into that later, but keep in mind that our ultimate goal is to find this solution set and express it elegantly using interval notation.

So, what's the general strategy for solving inequalities? It's actually quite similar to solving equations, with one key difference that we'll highlight later. We aim to isolate the variable (in this case, x) on one side of the inequality. We can do this by performing operations on both sides, just like we do with equations. Addition, subtraction, multiplication, and division – these are all our trusty tools. However, we need to be mindful of that one crucial difference: multiplying or dividing by a negative number. This is where things get a little tricky, and we'll explore why in detail when we encounter it in our problem. But for now, let's keep this in the back of our minds.

Step-by-Step Solution: Cracking the Code

Okay, let's get our hands dirty and solve the inequality $-10x + 2 \geq -11x + 13$ step by step. Remember, our mission is to isolate x on one side. The first thing we want to do is gather all the x terms together. We can achieve this by adding 11x to both sides of the inequality. This is a perfectly legal move, just like adding the same number to both sides of an equation. Why 11x? Because it will cancel out the -11x on the right side, bringing us closer to isolating x. Let's see how it looks:

$-10x + 2 + 11x \geq -11x + 13 + 11x$

Simplifying both sides, we get:

$x + 2 \geq 13$

Awesome! We've successfully moved the x terms to the left side. Now, we need to get rid of that pesky +2 that's hanging out with the x. To do this, we'll subtract 2 from both sides of the inequality. Again, this is a valid operation that maintains the balance of the inequality:

$x + 2 - 2 \geq 13 - 2$

Simplifying, we arrive at:

$x \geq 11$

Boom! We've done it! We've isolated x. This inequality, $x \geq 11$, tells us that the solution set consists of all values of x that are greater than or equal to 11. In other words, any number that is 11 or larger will satisfy the original inequality. This is a huge step, but we're not quite done yet. We need to express this solution set using interval notation.

Interval Notation: A Concise Language for Solutions

Now that we've found the solution – $x \geq 11$ – let's translate it into the elegant language of interval notation. Interval notation is a way of representing a continuous set of numbers using brackets and parentheses. The key is understanding what each symbol means.

• A square bracket [ ] indicates that the endpoint is included in the solution set. This is used when we have "greater than or equal to" ($\geq$) or "less than or equal to" ($\leq$) in our inequality.
• A parenthesis ( ) indicates that the endpoint is not included in the solution set. This is used when we have "greater than" ($>$) or "less than" ($<$) in our inequality.
• Infinity $\infty$ and negative infinity $-\infty$ are used to represent unbounded intervals, and they always get a parenthesis because we can never actually "reach" infinity.

So, how do we represent $x \geq 11$ in interval notation? Well, we know that our solution includes all numbers starting from 11 and going all the way up to infinity. Since 11 is included (due to the "equal to" part of the "greater than or equal to" symbol), we'll use a square bracket. And since infinity is never included, we'll use a parenthesis. Therefore, the interval notation for $x \geq 11$ is:

$[11, \infty)$

Let's break this down: The [11 tells us that the solution set starts at 11 and includes 11 itself. The , \infty) tells us that the solution set extends indefinitely to positive infinity. This concise notation beautifully captures the essence of our solution.

The Final Answer and Why It Matters

We've successfully navigated the world of inequalities, solved our problem, and expressed the solution in interval notation. The answer is $[11, \infty)$, which corresponds to option A in the multiple-choice options. But more than just getting the right answer, it's important to understand the process and the underlying concepts. Solving inequalities is a fundamental skill in mathematics, and it lays the groundwork for more advanced topics like calculus and linear programming.

Remember the key takeaways from our journey today:

• Inequalities deal with relationships where one side is greater than, less than, greater than or equal to, or less than or equal to the other side.
• The solution set of an inequality is the set of all values that make the inequality true.
• Interval notation is a concise way to represent a solution set using brackets and parentheses.
• Multiplying or dividing both sides of an inequality by a negative number requires flipping the inequality sign.

By mastering these concepts, you'll be well-equipped to tackle any inequality that comes your way. So, keep practicing, keep exploring, and keep unlocking the secrets of mathematics! You guys got this!