Solving Inequalities A Step-by-Step Guide To 1. 9x + 0.27 ≤ 7.23
Understanding the Problem
Alright, guys, let's dive into solving this inequality problem! We've got 1.9x + 0.27 ≤ 7.23, and our mission is to figure out what values of x make this statement true. Inequalities are super useful in math because they help us describe a range of possible solutions, not just one single answer. Think of it like setting a limit – we want to find all the x's that fit under or equal to a certain value.
The first thing we need to grasp is the concept of an inequality itself. Unlike an equation where we're looking for a specific value that makes both sides equal, an inequality deals with values that are either greater than, less than, greater than or equal to, or less than or equal to another value. This opens up a whole world of possibilities, which is pretty cool!
In our case, the ≤ symbol tells us we're looking for values of x that make the left side of the expression (1.9x + 0.27) less than or equal to the right side (7.23). So, we're not just hunting for one magic number; we're searching for a range of numbers that fit the bill. That's where the fun begins! To solve this, we’ll use techniques very similar to solving regular equations, but there's a tiny twist we need to keep in mind, which we'll get to later. We need to isolate x on one side of the inequality, and we'll do that by carefully undoing the operations that are being applied to it. Think of it like peeling back the layers of an onion – we'll start with the outermost layers and work our way in until we reveal the core, which is x itself.
Before we jump into the nitty-gritty, let's take a quick look at why inequalities are so important in the real world. You might be thinking, "Okay, this is a math problem, but where would I actually use this?" Well, inequalities pop up everywhere! Imagine you're budgeting your expenses – you might want to make sure your spending is less than or equal to your income. Or, if you're planning a road trip, you might want to calculate the maximum distance you can travel on a tank of gas. Inequalities help us set limits, find boundaries, and make informed decisions in all sorts of situations. They are truly powerful tools.
So, are you ready to become an inequality-solving pro? Let's grab our mathematical toolboxes and get started! We'll break down each step, explain the reasoning behind it, and make sure you feel confident tackling these types of problems. Remember, math is like a puzzle – and we're about to piece together the solution to this inequality.
Step-by-Step Solution
Okay, let's roll up our sleeves and get to work on solving the inequality 1.9x + 0.27 ≤ 7.23. The main goal here is to isolate x on one side of the inequality. We want to get x all by itself so we can see exactly what values it can take. To do this, we're going to use inverse operations – think of them as the undo buttons in math!
Step 1: Subtract 0.27 from both sides
The first thing we notice is that we have + 0.27 on the left side of the inequality. To get rid of this, we need to do the opposite operation, which is subtraction. We're going to subtract 0.27 from both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep things balanced. It's like a seesaw – if you take weight off one side, you need to take the same weight off the other to keep it level.
So, we have:
1.9x + 0.27 - 0.27 ≤ 7.23 - 0.27
This simplifies to:
1.9x ≤ 6.96
Awesome! We've made some progress. The 0.27 is gone from the left side, and we're one step closer to isolating x. Notice how we carefully lined up the subtraction and performed the calculation. Accuracy is key in math, guys, so always double-check your work!
Step 2: Divide both sides by 1.9
Now, we've got 1.9x ≤ 6.96. The x is being multiplied by 1.9, so to undo this multiplication, we need to divide. We're going to divide both sides of the inequality by 1.9. Again, it's all about keeping that balance. If we divide one side, we have to divide the other.
So, we have:
(1.9x) / 1.9 ≤ 6.96 / 1.9
This simplifies to:
x ≤ 3.6631578947...
Whoa, that's a bit of a long decimal! In most cases, we'll want to round this to a reasonable number of decimal places. Let's round it to two decimal places for now. So, we have:
x ≤ 3.66
Fantastic! We've done it. We've successfully isolated x. This means that any value of x that is less than or equal to 3.66 will satisfy the original inequality. That's a whole range of numbers! Think of all the possibilities: 3.66, 3.65, 0, -1, -100 – they all work!
Important Note: There's a crucial rule we need to remember when dealing with inequalities. If you multiply or divide both sides of an inequality by a negative number, you need to flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. Luckily, we didn't have to do that in this problem, but it's a super important thing to keep in mind for future problems.
Visualizing the Solution
Okay, we've solved the inequality 1.9x + 0.27 ≤ 7.23 and found that x ≤ 3.66. That's awesome! But sometimes, it helps to see what this solution actually means. One of the best ways to do that is by using a number line. Number lines are like visual maps of numbers, and they can make inequalities much easier to understand.
Drawing the Number Line
Imagine a straight line that stretches out in both directions forever. That's our number line! We'll mark zero somewhere in the middle as our starting point. To the right of zero are the positive numbers (1, 2, 3, and so on), and to the left of zero are the negative numbers (-1, -2, -3, and so on). Each point on the line represents a specific number.
Now, we need to represent our solution, x ≤ 3.66, on this number line. The first thing we'll do is find 3.66 on the line. It's going to be a little bit past 3, closer to 4, but not quite there. We'll mark that spot.
Using a Closed Circle or Open Circle
This is where things get interesting. Because our inequality is x ≤ 3.66 (less than or equal to), we're going to use a closed circle (or a filled-in dot) at 3.66. The closed circle tells us that 3.66 itself is included in the solution. If our inequality was just x < 3.66 (less than, but not equal to), we would use an open circle (a circle that's not filled in) to show that 3.66 is a boundary, but not actually part of the solution.
Shading the Solution
Next, we need to show all the other numbers that are part of the solution. Since x ≤ 3.66, we want all the numbers that are less than or equal to 3.66. On the number line, these are all the numbers to the left of 3.66. So, we're going to shade the number line to the left of our closed circle. This shaded region represents all the possible values of x that make our inequality true.
Interpreting the Number Line
When we look at our number line, we can clearly see the solution to the inequality. The closed circle at 3.66 tells us that 3.66 is included, and the shaded region stretching to the left tells us that all the numbers smaller than 3.66 are also solutions. It's like a visual confirmation of what we calculated algebraically!
Visualizing solutions on a number line is a powerful tool. It helps us understand the concept of inequalities in a more intuitive way. Instead of just seeing symbols and numbers, we can actually see the range of possible solutions. It's like turning an abstract idea into a concrete picture.
Real-World Applications
Okay, we've become pros at solving the inequality 1.9x + 0.27 ≤ 7.23. We know how to isolate x, we know how to interpret the solution, and we even know how to visualize it on a number line. But let's take it one step further, guys. Let's talk about why this stuff actually matters in the real world.
You might be thinking, "Okay, math is cool and all, but when am I ever going to use this in my day-to-day life?" Well, inequalities are surprisingly useful in a ton of different situations. They're not just abstract math concepts; they're tools that can help us make decisions, solve problems, and understand the world around us.
Budgeting and Finance
One of the most common places where inequalities come into play is in budgeting and finance. Let's say you're trying to save up for a new gadget. You might set a goal for how much money you want to save each month. That goal can be expressed as an inequality. For example, if you want to save at least $100 per month, you could write that as savings ≥ $100. This means your savings need to be greater than or equal to $100.
Or, let's say you have a certain amount of money to spend on groceries each week. You might want to make sure your spending is less than or equal to that amount. If you have $50 to spend, you could write that as spending ≤ $50. Inequalities help you set limits and make sure you're staying within your financial boundaries.
Constraints and Limitations
Inequalities are also used to represent constraints or limitations in various situations. Imagine you're planning a party and you have a limited amount of space. You can only invite a certain number of people. That limit can be expressed as an inequality. For instance, if you can only fit 20 people in your space, you could write that as number of guests ≤ 20.
Or, let's say you're baking cookies and you have a limited amount of flour. You need to make sure you don't use more flour than you have. If you have 5 cups of flour, you could write that as flour used ≤ 5 cups. Inequalities help us understand and work within the limits we face in the real world.
Science and Engineering
Inequalities are also crucial in science and engineering. For example, in physics, you might use inequalities to describe the range of possible values for a certain measurement. Or, in engineering, you might use them to design structures that can withstand certain loads or stresses. Inequalities help scientists and engineers make predictions, set safety limits, and ensure that things work properly.
Optimization Problems
Another important application of inequalities is in optimization problems. These are problems where you're trying to find the best solution, given certain constraints. For example, a company might want to maximize its profits while keeping its costs below a certain level. Inequalities are used to define the constraints and help find the optimal solution. These types of problems pop up in business, logistics, and many other fields.
Our Specific Example
Now, let's think about how our specific inequality, 1.9x + 0.27 ≤ 7.23, might relate to a real-world situation. Imagine x represents the number of hours you work at a part-time job. 1.9x could represent your earnings per hour (assuming you make $1.90 per hour). The 0.27 might be some fixed expense, like transportation costs to get to work. And the 7.23 could be the maximum amount of money you want to spend each day.
In this scenario, the inequality 1.9x + 0.27 ≤ 7.23 tells us the maximum number of hours you can work each day while still keeping your expenses below $7.23. Solving the inequality helps you make a practical decision about your work schedule. See? Math in action!
Conclusion
Alright, guys, we've reached the end of our inequality-solving adventure! We started with the problem 1.9x + 0.27 ≤ 7.23 and we've gone all the way from understanding the basics to seeing how it applies in the real world. That's a pretty awesome journey, if you ask me!
We've learned how to break down an inequality step-by-step, using inverse operations to isolate the variable. We've mastered the art of visualizing solutions on a number line, and we've even explored how inequalities show up in budgeting, finance, science, engineering, and everyday decision-making. It's like we've unlocked a secret code to understanding a big chunk of the world around us.
The key takeaway here is that inequalities aren't just abstract symbols and numbers. They're powerful tools that can help us make sense of constraints, set limits, and find the best solutions to real-world problems. Whether you're planning your budget, designing a building, or figuring out how many hours to work, inequalities can be your trusty sidekick.
Remember, the process we followed for solving 1.9x + 0.27 ≤ 7.23 can be applied to a wide range of inequalities. The core principles remain the same: use inverse operations, maintain balance on both sides, and pay attention to that all-important rule about flipping the inequality sign when multiplying or dividing by a negative number. With practice, you'll become an inequality-solving wizard in no time!
So, what's next? Well, now that you've got this solid foundation, you can tackle more complex inequalities, explore systems of inequalities, and even delve into the world of linear programming, where inequalities are used to optimize solutions in business and economics. The possibilities are endless!
Keep practicing, keep asking questions, and keep exploring the amazing world of mathematics. You've got this! And remember, math isn't just about finding the right answer; it's about developing problem-solving skills that you can use in every aspect of your life. So, go forth and conquer those inequalities – you've got the tools and the knowledge to succeed!
