Hey guys! Let's dive into solving the inequality -5x + 4 ≤ 24. This isn't as scary as it looks, I promise! We're going to break it down step by step, express the solution in set-builder notation, interval notation, and even graph it. By the end of this guide, you'll be a pro at tackling these types of problems. So, let’s get started and make math a little less intimidating and a lot more fun!
Understanding Inequalities
Before we jump into solving, let's quickly recap what inequalities are all about. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have a single solution (or a few discrete solutions), inequalities often have a range of solutions. This range can be expressed in a few different ways, which we'll explore later: set-builder notation, interval notation, and graphically. Understanding these different representations is key to mastering inequalities.
When you encounter an inequality, think of it as a question asking, "For what values of x does this statement hold true?" Our goal is to isolate x on one side of the inequality, just like we do with equations. However, there's one crucial difference: when we multiply or divide both sides of an inequality by a negative number, we need to flip the direction of the inequality sign. Keep this in mind, as it's a common mistake that can lead to incorrect solutions. It's also crucial to understand the properties of inequalities, which allow us to perform operations on both sides while maintaining the truth of the statement. For instance, adding or subtracting the same number from both sides doesn't change the inequality, similar to how it works with equations. This is a foundational concept, so make sure you’re comfortable with it before moving on.
Also, visualizing inequalities on a number line can be incredibly helpful. This visual representation gives you a clear picture of the solution set. A closed circle on the number line indicates that the endpoint is included in the solution (≤ or ≥), while an open circle indicates that the endpoint is not included (< or >). This visual aid can bridge the gap between abstract symbols and concrete understanding. So, with these fundamental concepts in mind, we're well-prepared to tackle our inequality head-on.
Step-by-Step Solution: -5x + 4 ≤ 24
Okay, let's tackle our inequality: -5x + 4 ≤ 24. Our mission is to isolate x on one side. Remember, we're solving for all the possible values of x that make this statement true. Think of it like solving a puzzle where x is the hidden piece we need to uncover. The steps are pretty straightforward, and you'll get the hang of it quickly. So, let's roll up our sleeves and get started!
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Isolate the term with x: To begin, we need to get the term with x (-5x) by itself on one side of the inequality. We can do this by subtracting 4 from both sides. This is like balancing a scale – whatever we do to one side, we must do to the other to keep it balanced.
-5x + 4 ≤ 24
-5x + 4 - 4 ≤ 24 - 4
-5x ≤ 20
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Solve for x: Now, we need to get x all alone. Since x is being multiplied by -5, we'll divide both sides by -5. Here's the crucial part: because we're dividing by a negative number, we must flip the direction of the inequality sign. This is a key rule to remember when working with inequalities. It's like crossing over into another dimension where the rules are slightly different. If you forget this step, your answer will be incorrect!
-5x ≤ 20
(-5x) / -5 ≥ 20 / -5
x ≥ -4
And there we have it! The solution to the inequality is x ≥ -4. This means any value of x that is greater than or equal to -4 will satisfy the original inequality. Isn’t that neat? We've transformed a seemingly complex problem into a clear and concise solution. This is the power of algebra at work, and you're now one step closer to mastering it.
Expressing the Solution
Now that we've solved the inequality, let's express the solution in different notations: set-builder and interval notation. These are just fancy ways of writing the same thing, but they each have their own advantages and are used in different contexts. Think of them as different languages for expressing the same mathematical idea. Learning to translate between these notations is like becoming multilingual in the world of mathematics. So, let's dive in and see how it's done!
Set-Builder Notation
Set-builder notation is a way of defining a set by specifying the properties its elements must satisfy. It's like giving a detailed description of the members of a club. In our case, we want to describe the set of all x values that are greater than or equal to -4. The general form of set-builder notation is { x | condition }, which is read as "the set of all x such that the condition is true." For our inequality, the set-builder notation looks like this:
{ x | x ≥ -4 }
This is read as "the set of all x such that x is greater than or equal to -4." See how neatly it encapsulates our solution? It's a concise and precise way of defining the solution set. Using set-builder notation is like writing a precise definition in a dictionary – it leaves no room for ambiguity.
Interval Notation
Interval notation is another way to represent a set of numbers, but it uses intervals on the number line. It's like giving a range within which the solutions lie. We use brackets [ ] to include the endpoints of the interval and parentheses ( ) to exclude them. Since our solution includes -4 (because x can be equal to -4), we'll use a bracket. And since the solution extends infinitely to the right (all numbers greater than -4), we'll use the infinity symbol (∞) and a parenthesis (since infinity is not a specific number and cannot be included). So, the interval notation for our solution is:
[-4, ∞)
This means the solution includes all numbers from -4 up to infinity, including -4 itself. Interval notation is incredibly useful because it provides a quick visual sense of the solution set on the number line. It's like reading a map where the solution is a specific route or area. Becoming fluent in interval notation is a valuable skill for understanding and communicating mathematical concepts.
Graphing the Solution
Finally, let's visualize the solution on a number line. This is a great way to get a clear picture of what our solution x ≥ -4 really means. Graphing the solution turns the abstract into something tangible and easy to grasp. Think of it like drawing a picture that illustrates the solution in a visually compelling way.
To graph the solution, we'll draw a number line and mark -4 on it. Since our solution includes -4 (x ≥ -4), we'll use a closed circle (or a solid dot) at -4. This indicates that -4 is part of the solution. Then, since x can be any number greater than -4, we'll draw an arrow extending to the right from -4. This arrow represents all the numbers greater than -4 that satisfy our inequality.
[Imagine a number line here with a closed circle at -4 and an arrow extending to the right.]
The graph provides an intuitive representation of the solution set. It allows you to quickly see the range of values that satisfy the inequality. Graphing the solution is like seeing the answer in living color, making it easier to understand and remember. So, by combining the algebraic solution with its graphical representation, we gain a comprehensive understanding of the inequality and its solutions.
Conclusion
Great job, guys! We've successfully solved the inequality -5x + 4 ≤ 24, expressed the solution in set-builder notation { x | x ≥ -4 }, interval notation [-4, ∞), and graphed it on a number line. We took a problem, broke it down step by step, and conquered it. That's the spirit of mathematics – taking on challenges and finding solutions!
Remember, the key to mastering inequalities (and any math topic) is practice. The more you work through problems, the more comfortable you'll become with the concepts and techniques. So, don't be afraid to tackle more inequalities, experiment with different notations, and visualize your solutions on a number line. Keep practicing, and you'll become a math whiz in no time!
Keep up the awesome work, and I'll see you in the next math adventure! Remember, math isn't just about numbers and symbols; it's about problem-solving, critical thinking, and the thrill of discovery. So, embrace the challenge, and let's keep learning and growing together. You've got this!