Decoding The Inequality 67z + 8.9 ≤ -302 A Comprehensive Guide
Hey guys! Today, we're diving into the world of mathematical inequalities and translating them into plain English. Specifically, we're going to break down the sentence represented by the inequality $67z + 8.9 \leq -302$. This might seem daunting at first, but trust me, it's like deciphering a secret code, and once you get the hang of it, it's super cool! So, grab your mental magnifying glasses, and let's get started!
Understanding the Inequality
To truly understand the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two, we need to dissect each part of it. Think of it like a sentence, where each symbol and number plays a specific role. The variable 'z' represents an unknown number, which is the mystery we're trying to solve. The number 67 is the coefficient, meaning it's being multiplied by our unknown 'z'. Then we have '+ 8.9', which indicates that we're adding eight and nine-tenths to the product of 67 and 'z'. And finally, the symbol '$\leq$' is the star of the show, telling us that the entire expression on the left side is "less than or equal to" the number on the right, which is -302. So, in essence, the inequality is a statement that compares two quantities, saying that one is either smaller than or possibly the same as the other.
Now, let's break down the mathematical symbols into everyday language. When we say "67z", we're essentially saying "sixty-seven times a number." The plus sign, '+', is straightforward; it means we're adding something. The '8.9' is simply "eight and nine-tenths." The tricky part might be the '$\leq$' symbol. This symbol combines two concepts: "less than" and "equal to." So, when we read "$\leq$", we say "less than or equal to" or, equivalently, "is not greater than." This is a crucial distinction, because it allows for the possibility that the two sides of the inequality could be equal. Lastly, "-302" is just "negative three hundred two."
By understanding each component of the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two, we can start to see how it translates into a full sentence. The key is to connect the mathematical symbols and operations with their corresponding words. Think of it like building a bridge between two languages – math and English. Once we have a solid grasp of the individual pieces, we can put them together to form a clear and accurate sentence. Remember, the goal is to capture the precise meaning of the inequality in a way that's easy to understand for anyone, even if they're not fluent in math-speak!
Translating the Inequality into a Sentence
Now comes the fun part: translating the inequality $67z + 8.9 \leq -302$ into a sentence. We've already dissected the individual components, so let's piece them together like a puzzle. Remember, we're aiming for a sentence that accurately reflects the mathematical relationship expressed in the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two. There are several ways we can phrase this, but the goal is to be both precise and clear.
Starting with the left side of the inequality, "67z" translates to "sixty-seven times a number." The "z" represents our unknown number, and the 67 is multiplying it. Next, we have "+ 8.9," which simply means "plus eight and nine-tenths." So far, we have "sixty-seven times a number, plus eight and nine-tenths." Now for the crucial part: the inequality symbol "$\leq$." As we discussed earlier, this means "less than or equal to," which can also be phrased as "is not greater than." Finally, we have "-302," which is "negative three hundred two." Putting it all together, we get the sentence:
"Sixty-seven times a number, plus eight and nine-tenths, is less than or equal to negative three hundred two."
Alternatively, we can use the "is not greater than" phrasing for the inequality symbol: "Sixty-seven times a number, plus eight and nine-tenths, is not greater than negative three hundred two." Both sentences accurately represent the mathematical relationship expressed in the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two. The choice between them often comes down to personal preference or the specific context. The key is to ensure that the sentence captures the meaning of "less than or equal to," acknowledging that the left side can be either smaller than or equal to the right side.
It's also important to note what the sentence doesn't say. It doesn't say that the expression is equal to negative three hundred two, only that it's no greater than it. This subtle but significant distinction is what makes inequalities so powerful in mathematics. They allow us to express a range of possible values, rather than just a single, fixed value. So, the next time you see an inequality, remember that it's telling a story about the relationship between two quantities, and translating it into a sentence is like revealing the plot!
Analyzing the Given Options
Okay, now that we've translated the inequality $67z + 8.9 \leq -302$ into a sentence ourselves, let's put our skills to the test and analyze the given options. This is where we see if we've truly understood the nuances of the inequality and can distinguish the correct representation from the incorrect ones. Remember, the devil's in the details, and even a small difference in wording can completely change the meaning of the sentence.
Let's revisit the options:
A. Sixty-seven times a number is less than negative three hundred two. B. Sixty-seven times a number, plus eight and nine tenths, is not greater than negative three hundred two.
Option A sounds similar, but let's break it down and compare it to the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two. This sentence translates to the inequality $67z < -302$. Notice anything missing? It doesn't include the "+ 8.9" part. This means it's only representing a portion of the original inequality. While it captures the "less than" relationship, it completely ignores the addition of eight and nine-tenths. Therefore, Option A is not a complete and accurate representation of the given inequality. It's like telling only half the story, and in mathematics, accuracy is key.
Now, let's examine Option B: "Sixty-seven times a number, plus eight and nine tenths, is not greater than negative three hundred two." This sentence sounds very familiar, doesn't it? That's because it's precisely the sentence we arrived at when we translated the inequality ourselves! It includes all the essential components: "sixty-seven times a number" (67z), "plus eight and nine-tenths" (+ 8.9), and "is not greater than" ($\leq$), which is equivalent to "less than or equal to." This sentence captures the entire essence of the inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two, expressing the relationship between the expression on the left and the number on the right. So, Option B is looking like a strong contender!
By carefully analyzing each option and comparing it to our own translation, we can confidently identify the sentence that accurately represents the inequality. This process highlights the importance of understanding not only the individual components of an inequality but also how they connect to form a complete mathematical statement. And remember, guys, practice makes perfect! The more you translate inequalities into sentences and vice versa, the easier it will become to spot the correct representation.
The Correct Answer: Option B
After our detailed analysis, it's clear that Option B, "Sixty-seven times a number, plus eight and nine tenths, is not greater than negative three hundred two," is the sentence that accurately represents the inequality $67z + 8.9 \leq -302$. We arrived at this conclusion by dissecting the inequality, translating each component into plain English, and then comparing our translation with the given options. Option B perfectly captures the mathematical relationship expressed in the inequality, including the multiplication, addition, and the "less than or equal to" relationship.
Let's recap why Option B is the correct answer. It includes the phrase "sixty-seven times a number," which corresponds to the 67z term. It also incorporates "plus eight and nine-tenths," representing the + 8.9. And most importantly, it uses the phrase "is not greater than," which is the key to understanding the "$\leq$" symbol. This symbol means "less than or equal to," and "is not greater than" is an equivalent way of expressing the same concept. By using this phrasing, Option B acknowledges that the expression on the left side of the inequality can be either less than or equal to negative three hundred two.
Option A, on the other hand, falls short because it omits the "+ 8.9" part of the inequality. It only focuses on the "67z" and the "less than" relationship, neglecting a crucial component of the expression. This omission makes Option A an incomplete and therefore inaccurate representation of the original inequality sixty-seven z plus eight point nine is less than or equal to negative three hundred two. In mathematics, precision is paramount, and we need to ensure that our sentences capture the entire mathematical statement.
So, hats off to Option B for being the clear winner! This exercise demonstrates the importance of careful translation and attention to detail when working with mathematical expressions. It's not enough to just get the general idea; we need to be precise in our language to accurately convey the mathematical meaning. And remember, guys, even seemingly small differences in wording can have a big impact on the overall meaning of a sentence.
Key Takeaways and Tips
Alright, guys, we've successfully decoded the inequality $67z + 8.9 \leq -302$ and found its corresponding sentence. But before we wrap up, let's highlight some key takeaways and tips that will help you tackle similar problems in the future. Translating mathematical expressions into sentences (and vice versa) is a fundamental skill in algebra and beyond, so mastering these concepts is super important.
First and foremost, understand the symbols. Mathematical symbols are like a secret language, and you need to know what they mean to decipher the message. We've already discussed the "$\leq$" symbol, but let's recap: it means "less than or equal to," which is the same as "is not greater than." Other important symbols to remember include ">" (greater than), "$\geq$" (greater than or equal to), "<" (less than), and "=" (equal to). Knowing these symbols and their corresponding phrases is the first step in accurate translation.
Break it down into smaller parts. Complex expressions can seem overwhelming at first, but if you break them down into smaller, more manageable pieces, they become much easier to handle. Identify the variables, coefficients, operations, and inequality symbols, and translate each one individually. Then, piece them together to form the complete sentence. This "divide and conquer" approach is a powerful strategy for solving many mathematical problems.
Pay attention to the order of operations. The order in which operations are performed is crucial in mathematics. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? This order dictates the sequence in which you should perform operations when evaluating an expression. When translating an expression, make sure your sentence reflects the correct order of operations. For example, "sixty-seven times a number, plus eight and nine-tenths" clearly indicates that the multiplication should be performed before the addition.
Practice, practice, practice! Like any skill, translating mathematical expressions takes practice. The more you do it, the more comfortable and confident you'll become. Work through examples in your textbook, online, or create your own. Challenge yourself to translate both inequalities and equations, and try translating sentences back into mathematical expressions. The more you practice, the better you'll get at spotting the key components and constructing accurate translations.
Don't be afraid to rephrase. There's often more than one way to express the same mathematical relationship in words. If your initial translation doesn't sound quite right, try rephrasing it. Experiment with different word choices and sentence structures until you find a version that accurately captures the meaning of the expression. Remember, clarity is key, so aim for a sentence that is both precise and easy to understand.
By following these tips and practicing regularly, you'll become a pro at translating mathematical expressions into sentences and vice versa. So, keep up the great work, guys, and remember that math is just another language waiting to be learned!
So, there you have it, guys! We've successfully navigated the world of inequalities, deciphered the meaning of $67z + 8.9 \leq -302$, and translated it into the sentence "Sixty-seven times a number, plus eight and nine tenths, is not greater than negative three hundred two." We've also explored some key strategies for tackling similar problems, from understanding mathematical symbols to breaking down complex expressions into smaller parts. Remember, translating mathematical expressions is like learning a new language, and with practice and the right tools, you can become fluent in math-speak!
This exercise highlights the importance of precision and attention to detail in mathematics. Even small differences in wording can significantly alter the meaning of a sentence, so it's crucial to choose your words carefully. By understanding the nuances of mathematical symbols and operations, you can accurately convey complex relationships and solve challenging problems. And remember, math is not just about numbers and equations; it's about logical thinking and clear communication.
The ability to translate mathematical expressions into sentences is a valuable skill that extends beyond the classroom. It helps you develop your critical thinking abilities, improve your communication skills, and gain a deeper understanding of the world around you. Math is everywhere, from calculating grocery bills to understanding scientific research, and being able to translate mathematical concepts into everyday language makes it more accessible and relevant.
So, keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is full of fascinating concepts and rewarding challenges, and with a little effort and the right mindset, you can unlock its secrets. And remember, guys, learning math can be fun! So embrace the challenge, enjoy the process, and celebrate your successes along the way. You've got this! Now go out there and conquer the mathematical world, one sentence at a time!
