Solving Inequalities: Finding The Solution Set

Hey everyone! Today, we're diving into the world of inequalities, specifically tackling the question: Which interval represents the solution set for the inequality $0.35x - 4.8 < 5.2 - 0.9x$? Sounds fun, right? Don't worry, it's not as scary as it looks. We'll break it down step-by-step, making sure you grasp every detail. Think of it like a treasure hunt, and we're after the hidden interval that satisfies the inequality. Let's get started and unravel this mathematical mystery together! We will explore a simple algebraic inequality, providing a detailed, step-by-step solution to help you understand the process. The goal is not just to find the answer but to understand why the answer is what it is. This is crucial for building a strong foundation in algebra. We will analyze each step carefully, clarifying any potential points of confusion. By the end of this, you should be able to confidently solve similar inequalities. Now, let's look at the given options:

A. $(-\\infty, -8)$

B. $(-\\infty, 8)$

C. $(-8, \\infty)$

D. $(8, \\infty)$

Our task is to find the correct interval among these options. Ready? Let's go!

Understanding the Inequality

Alright, before we jump into solving the inequality $0.35x - 4.8 < 5.2 - 0.9x$, let's make sure we're all on the same page. Inequalities are like equations, but instead of an equals sign (=), they use symbols like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). These symbols tell us that one side of the inequality is not exactly equal to the other; instead, it's either smaller or larger.

In our case, the inequality $0.35x - 4.8 < 5.2 - 0.9x$ is asking us to find all the values of 'x' for which the expression on the left side ($0.35x - 4.8$) is less than the expression on the right side ($5.2 - 0.9x$). Think of it as a balance scale. We're trying to figure out what values of 'x' would make the left side of the scale lighter than the right side.

So, why do we care about inequalities? Well, they're super important in real-world scenarios! For example, they can help you determine how much you can spend without exceeding a budget, or find the range of values that will keep a machine running safely. Now, let's get down to business and solve this inequality. Remember, our goal is to isolate 'x' on one side of the inequality and determine its possible values, which will give us our solution set. Let's do this!

Step-by-Step Solution

Okay, buckle up, because we're about to solve this inequality step-by-step. First, let's write down the inequality we need to solve: $0.35x - 4.8 < 5.2 - 0.9x$. Our primary goal is to isolate the variable 'x' on one side of the inequality. To do this, we'll use some algebraic manipulations, similar to solving equations, but with a slight twist. Remember, when we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. Let's start by getting all the 'x' terms on one side of the inequality. We can do this by adding $0.9x$ to both sides. This cancels out the $-0.9x$ on the right side. Here’s what it looks like:

$0.35x - 4.8 + 0.9x < 5.2 - 0.9x + 0.9x$

This simplifies to:

$1.25x - 4.8 < 5.2$

Next, we need to isolate the 'x' term. We can do this by adding $4.8$ to both sides of the inequality. This cancels out the $-4.8$ on the left side. So we have:

$1.25x - 4.8 + 4.8 < 5.2 + 4.8$

Which simplifies to:

$1.25x < 10$

Now, to solve for 'x', we divide both sides of the inequality by $1.25$. Since we're dividing by a positive number, we don't need to flip the inequality sign.

$1.25x / 1.25 < 10 / 1.25$

This simplifies to:

$x < 8$

And there we have it! We have successfully isolated 'x'. This tells us that the solution to the inequality is all values of 'x' that are less than 8. Let's see how we can express this as an interval.

Expressing the Solution as an Interval

Great job on solving the inequality, everyone! We found that the solution is $x < 8$. Now, how do we represent this solution in interval notation? Interval notation is a way to express a set of numbers that includes all the numbers between two endpoints. In our case, the solution set includes all real numbers less than 8. The symbol for infinity is ∞. If we have an inequality such as x < 8, then the interval starts from negative infinity and goes up to, but does not include, 8. So, we'll use a parenthesis to indicate that 8 is not included in the solution set. Therefore, the interval is $(-\\infty, 8)$. Let's break down the components of this interval notation to make sure it's crystal clear.

1. (-\infty: This part indicates that the interval starts from negative infinity. It means that the solution includes all real numbers that are less than 8, going all the way down to negative infinity. Negative infinity is always represented with a parenthesis because it is not a specific number.
2. , 8): This part indicates that the interval goes up to 8. The parenthesis around 8 shows that 8 itself is not included in the solution. We use a parenthesis because the inequality is '<', which means 'less than', not 'less than or equal to'. If the inequality had been 'x ≤ 8', we would have used a bracket '[8' to show that 8 is included in the solution. The correct interval notation is thus $(-\\infty, 8)$. Remember, when we use a parenthesis, we are saying that the endpoint is not included in the interval. Using a bracket, on the other hand, means the endpoint is included. Now that we've found our interval, let's go back and match our solution to the multiple-choice options we were given earlier.

Matching the Solution to the Options

Alright, we've done all the hard work – solving the inequality and expressing the solution in interval notation. Now comes the easy part: matching our answer to the given options. We found that the solution to the inequality $0.35x - 4.8 < 5.2 - 0.9x$ is $x < 8$, which in interval notation is $(-\\infty, 8)$. Now let's revisit the options:

A. $(-\\infty, -8)$

B. $(-\\infty, 8)$

C. $(-8, \\infty)$

D. $(8, \\infty)$

By comparing our solution $(-\\infty, 8)$ with the options provided, we can easily see that option B matches our solution perfectly! Congratulations, you did it! Understanding how to solve inequalities and represent their solutions is a fundamental skill in algebra. Keep practicing, and you'll become a pro in no time! So, to recap, we first solved the inequality to find the range of 'x' values that satisfy it. Then, we expressed this range using interval notation. Finally, we matched this solution to the provided options. This process not only helps you find the correct answer, but it also reinforces your understanding of the concepts involved.

Conclusion

So there you have it, folks! We've successfully solved the inequality $0.35x - 4.8 < 5.2 - 0.9x$ and found that the solution set is represented by the interval $(-\\infty, 8)$. Remember, solving inequalities is a fundamental skill in algebra, and with practice, you'll become a pro at it. Keep an eye out for more math problems, and don't hesitate to give them a try. You've got this! Remember, understanding the why behind each step is as important as getting the right answer. Keep up the excellent work, and always strive to deepen your understanding of these concepts. Math can be fun and rewarding, and with consistent effort, you'll achieve great things! Keep practicing, and you'll be solving inequalities like a boss in no time. Thanks for joining me today. See you next time, and happy solving!