Solving Inequalities Three Times Two Less Than A Number

Hey guys! Let's dive into a fun mathematical puzzle today. We're going to break down a word problem, translate it into an inequality, and then solve it to find the numbers that fit the description. It might sound a bit intimidating at first, but trust me, we'll take it step-by-step, and you'll see it's not as scary as it seems! So, grab your thinking caps, and let's get started!

Unraveling the Problem Statement

The core of our challenge lies in understanding this statement: "Three times two less than a number is greater than or equal to five times the number." Sounds like a tongue twister, right? But let's dissect it piece by piece. The most crucial part is identifying the unknown – the "number." In mathematics, we often represent unknowns with variables, and in this case, we're told to let "n" represent our mystery number. This is our first step in translating the words into a mathematical expression. Think of 'n' as a placeholder, a box that we need to fill with the correct numbers that make the statement true.

Now, let's break down the other parts of the sentence. "Two less than a number" translates directly to "n - 2." This is because we're taking 2 away from our unknown number 'n.' Next, we have "three times two less than a number," which means we're multiplying the entire expression "n - 2" by 3. This gives us "3(n - 2)." Remember, the parentheses are super important here! They tell us to perform the subtraction first and then multiply the result by 3. It's like following a recipe – you need to do the steps in the right order to get the desired outcome. If we didn't use parentheses, we'd be multiplying only the 'n' by 3, which would change the whole meaning of the expression.

On the other side of the comparison, we have "five times the number," which simply means "5n." This is straightforward – we're multiplying our unknown number 'n' by 5. Finally, we have the phrase "is greater than or equal to," which is a mathematical way of saying that one value is either bigger than the other or the same as the other. In mathematical symbols, we represent this with the "≥" symbol. This symbol is like a combination of "greater than" (>) and "equal to" (=), indicating that both conditions are acceptable. Think of it as a flexible rule – the number can be larger, or it can be exactly the same, and it still fits the criteria.

Putting it all together, we can translate the entire sentence into a mathematical inequality: 3(n - 2) ≥ 5n. This inequality is the heart of our problem. It's a concise way of expressing the relationship described in the words, and it's what we'll use to find the numbers that satisfy the condition. Now, the real fun begins – solving this inequality to uncover the possible values of 'n'! This is where our algebraic skills come into play, and we'll use techniques like distribution, combining like terms, and isolating the variable to find our solution. Stay tuned, guys, we're getting closer to cracking this number puzzle!

Choosing the Right Inequality

Now that we've meticulously broken down the problem statement, the next crucial step is to correctly represent it as a mathematical inequality. This is where precision is key, guys, because the wrong inequality will lead us down the wrong path! Remember, the statement we're working with is: "Three times two less than a number is greater than or equal to five times the number." We've already established that "n" represents our unknown number.

Let's revisit the pieces we've already translated. "Two less than a number" is "n - 2." "Three times two less than a number" is "3(n - 2)." And "five times the number" is "5n." The most important part, the connective tissue of the inequality, is "is greater than or equal to," which, as we discussed, is represented by the symbol "≥." This symbol is the bridge that connects the two sides of our inequality, telling us how they relate to each other. It's not just saying that one side is bigger; it's also including the possibility that they could be exactly the same.

So, when we combine all these elements, we arrive at the inequality: 3(n - 2) ≥ 5n. This is the algebraic representation of our word problem, the precise mathematical sentence that captures the essence of the statement. It's like translating a sentence from one language to another – we're taking the words and converting them into a symbolic form that we can manipulate and solve. But why is choosing the correct inequality so vital? Well, think of it like this: if you have a map with the wrong directions, you're not going to reach your destination, no matter how well you follow the map. Similarly, if we start with the wrong inequality, our solution will be incorrect, even if our algebraic steps are flawless. This is why careful attention to detail in this initial translation phase is absolutely essential.

Imagine if we had mistakenly chosen the "less than or equal to" symbol (≤) instead of "greater than or equal to" (≥). This seemingly small change would completely reverse the relationship between the two sides of the inequality, leading us to a completely different set of solutions. It's like saying that the temperature is below freezing when it's actually a warm summer day – the information is fundamentally wrong. Similarly, if we forgot the parentheses around "n - 2," we would be changing the order of operations, which would also distort the meaning of the expression. Parentheses are like traffic signals in mathematics – they tell us when to stop, go, and which way to turn.

Therefore, selecting the correct inequality is not just a minor detail; it's the foundation upon which our entire solution rests. It's about accurately capturing the relationships described in the word problem and setting ourselves up for success in the next steps. So, with our inequality 3(n - 2) ≥ 5n firmly in place, we're ready to move on to the exciting part: actually solving for 'n' and finding the numbers that make this statement true! Let's keep going, guys; we're on the verge of unlocking the solution!

Solving the Inequality Step-by-Step

Alright, guys, we've successfully translated the word problem into the inequality 3(n - 2) ≥ 5n. Now comes the fun part – actually solving for 'n'! This is where we get to put our algebra skills to the test and unravel the mystery of which numbers satisfy this condition. Don't worry; we'll take it step-by-step, making sure each move is clear and logical.

The first thing we need to do is simplify the left side of the inequality. We have 3(n - 2), which means we need to distribute the 3 across the terms inside the parentheses. Remember the distributive property? It's like sharing the 3 with both the 'n' and the '-2.' So, 3 times 'n' is 3n, and 3 times -2 is -6. This gives us 3n - 6. Now, our inequality looks like this: 3n - 6 ≥ 5n. We've taken the first step towards simplification, and things are already looking a little less cluttered!

Next, we want to get all the 'n' terms on one side of the inequality and the constant terms on the other side. It's like sorting laundry – we want to group similar items together. To do this, let's subtract 3n from both sides of the inequality. Why subtract 3n? Because it will eliminate the 'n' term from the left side. Remember, whatever we do to one side of the inequality, we must do to the other to maintain the balance. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. Subtracting 3n from both sides gives us: 3n - 6 - 3n ≥ 5n - 3n, which simplifies to -6 ≥ 2n. We're making good progress! The 'n' terms are now neatly grouped on the right side.

Now, we need to isolate 'n' completely. It's like freeing the variable from its numerical shackles! To do this, we need to get rid of the 2 that's multiplying 'n.' How do we undo multiplication? We divide! So, we'll divide both sides of the inequality by 2. This gives us -6 / 2 ≥ 2n / 2, which simplifies to -3 ≥ n. We're almost there! We've got 'n' practically all by itself.

However, there's one tiny detail we need to address. It's a little mathematical quirk that can sometimes trip people up. We have -3 ≥ n, which means -3 is greater than or equal to 'n.' But it's often easier to understand the solution if we have 'n' on the left side. So, let's flip the inequality around. But here's the crucial part: when we flip an inequality, we also need to flip the inequality sign! It's like looking at a reflection in a mirror – everything is reversed. So, -3 ≥ n becomes n ≤ -3. And there we have it! Our final solution.

Interpreting the Solution and Finding the Numbers

Okay, guys, after all our hard work, we've arrived at the solution: n ≤ -3. But what does this actually mean? It's not just a jumble of symbols; it's a statement about the numbers that satisfy the original condition. This inequality is telling us that 'n' can be any number that is less than or equal to -3. Think of it as a range of possibilities, a whole set of numbers that fit the bill.

To truly understand the solution, let's break it down. The "≤" symbol means "less than or equal to." So, 'n' can be equal to -3, or it can be any number smaller than -3. This includes numbers like -4, -5, -6, and so on, stretching infinitely in the negative direction. It's like a one-way street extending towards negative infinity.

Let's consider a few examples to make this concrete. If n = -3, then the original inequality becomes 3(-3 - 2) ≥ 5(-3), which simplifies to 3(-5) ≥ -15, and further to -15 ≥ -15. This is true, since -15 is indeed equal to -15. So, -3 is part of our solution set.

Now, let's try n = -4. The inequality becomes 3(-4 - 2) ≥ 5(-4), which simplifies to 3(-6) ≥ -20, and further to -18 ≥ -20. This is also true, since -18 is greater than -20. So, -4 is also a valid solution.

But what about a number that's not less than or equal to -3, like, say, 0? If n = 0, the inequality becomes 3(0 - 2) ≥ 5(0), which simplifies to 3(-2) ≥ 0, and further to -6 ≥ 0. This is false, since -6 is not greater than or equal to 0. This confirms that 0 is not part of our solution set.

So, we've found a whole range of numbers that make the original statement true. Any number that is -3 or smaller will satisfy the condition "three times two less than a number is greater than or equal to five times the number." This is a powerful result! We've taken a word problem, translated it into a mathematical inequality, solved the inequality, and then interpreted the solution to find a set of numbers that fit the given criteria. It's like being a mathematical detective, following the clues and uncovering the answer. Guys, we've successfully cracked this number puzzle! Give yourselves a pat on the back; you've earned it!

So there you have it, guys! We've successfully navigated the world of inequalities, taking a word problem and transforming it into a solvable mathematical statement. We've seen how crucial it is to break down complex sentences, identify the unknowns, and translate the relationships between them into algebraic expressions. We've also learned the importance of choosing the correct inequality symbol and the steps involved in solving inequalities, including distribution, combining like terms, and isolating the variable. But perhaps the most important takeaway is the ability to interpret the solution in the context of the original problem.

This exercise demonstrates the power of mathematical translation – the ability to take real-world scenarios and express them in the precise language of mathematics. This skill is not just useful for solving textbook problems; it's a fundamental tool for problem-solving in many areas of life, from science and engineering to finance and economics. By learning to think mathematically, we can approach challenges with a logical and systematic mindset, breaking them down into manageable parts and finding solutions that might not be immediately obvious.

Remember, guys, mathematics isn't just about memorizing formulas and procedures; it's about developing a way of thinking, a way of approaching problems with clarity and precision. And like any skill, it takes practice. The more you work with these concepts, the more comfortable and confident you'll become. So, keep exploring, keep questioning, and keep challenging yourselves with new mathematical puzzles. The world of numbers and equations is full of fascinating discoveries just waiting to be made!