Solve: Three Times Two Less Than A Number

Hey guys, let's dive into a fun mathematical puzzle! We're going to break down the statement "Three times two less than a number is greater than or equal to five times the number." Sounds like a mouthful, right? But don't worry, we'll make it super clear and easy to understand. Our mission is to find all the numbers that fit this description. So, buckle up, and let's get started!

Unraveling the Mystery: Translating Words into Math

First things first, we need to translate this wordy statement into a mathematical inequality. This is like learning a new language, where we convert everyday words into symbols and operations. The key to cracking this code lies in carefully dissecting each phrase and understanding its mathematical equivalent. Let’s take it bit by bit:

• "A number": This is our mystery number, the one we're trying to find. In algebra, we often represent unknowns with variables, and in this case, the problem gives us a head start by telling us to use n. So, n is our secret number.
• "Two less than a number": This means we're taking our number n and subtracting 2 from it. Mathematically, this is written as n - 2.
• "Three times two less than a number": Now we're multiplying the previous expression (n - 2) by 3. This gives us 3(n - 2). Remember, the parentheses are crucial here because they ensure we're multiplying the entire quantity n - 2 by 3, not just the 2.
• "Five times the number": This one's straightforward. We're multiplying our number n by 5, which is written as 5n.
• "Is greater than or equal to": This is our inequality symbol. It tells us that one quantity is either bigger than or the same as another. The symbol for this is ≥.

Now, let's put it all together. "Three times two less than a number is greater than or equal to five times the number" translates to the inequality 3(n - 2) ≥ 5n. See? It's like we've deciphered a secret message!

Choosing the Right Inequality: Why This Matters

Selecting the correct inequality is super important because it sets the stage for solving the problem. If we misinterpret the words and choose the wrong symbol, our entire solution will be off. Think of it like following a recipe – if you mix up the ingredients, the final dish won't taste right. In our case, the phrase "greater than or equal to" leaves no room for ambiguity; it clearly points to the ≥ symbol. Had the statement said "greater than," we would have used >, or if it said "less than or equal to," we'd opt for ≤. The subtle differences in these phrases make a big difference in the mathematical world!

Cracking the Code: Solving the Inequality

Alright, we've got our inequality: 3(n - 2) ≥ 5n. Now comes the fun part – solving for n! This is like a detective solving a case, where we use mathematical tools and techniques to isolate our unknown number and reveal its true value. The goal is to get n all by itself on one side of the inequality, so we know exactly what numbers satisfy the condition.

Step-by-Step Solution: Unveiling the Numbers

Let's break down the solving process into simple steps:

1. Distribute: We need to get rid of the parentheses first. Remember the distributive property? It says that a(b + c) = ab + ac. So, we multiply the 3 by both terms inside the parentheses: 3 * n = 3n and 3 * -2 = -6. This gives us 3n - 6 ≥ 5n.
2. Gather the 'n' terms: Our next goal is to get all the terms with n on the same side of the inequality. To do this, we can subtract 3n from both sides. This keeps the inequality balanced, just like a scale. Subtracting 3n from both sides gives us -6 ≥ 5n - 3n, which simplifies to -6 ≥ 2n.
3. Isolate 'n': Now we want to get n all by itself. It's currently being multiplied by 2, so we need to do the opposite operation: divide both sides by 2. This gives us -6 / 2 ≥ n, which simplifies to -3 ≥ n.
4. Flip it around (optional, but helpful): Some people find it easier to read the inequality with the variable on the left. We can flip the inequality around as long as we also flip the direction of the inequality symbol. So, -3 ≥ n is the same as n ≤ -3.

We've done it! We've solved for n. Our solution is n ≤ -3. This means that any number that is less than or equal to -3 will satisfy our original condition. That's a whole range of numbers! Think of it like a club where only numbers -3 and smaller are allowed to join.

The Critical Flip: Dividing by a Negative

Now, let's imagine a slightly different scenario. Suppose, during our solving process, we encountered a step where we needed to divide both sides of the inequality by a negative number. This is a crucial point because dividing (or multiplying) an inequality by a negative number requires a special action: we must flip the direction of the inequality symbol. Why? Because multiplying or dividing by a negative number changes the order of numbers on the number line. For example, 2 is greater than 1, but if we multiply both by -1, we get -2 and -1, and now -2 is less than -1. So, the direction of the relationship flips.

If we had an inequality like -2n > 4, to solve for n, we would divide both sides by -2, and the inequality would become n < -2. This flip is super important to remember, or else we'll end up with the wrong solution set!

The Grand Finale: Interpreting the Solution

We've arrived at the final stage of our mathematical journey: interpreting what our solution n ≤ -3 actually means. It's not enough just to get the answer; we need to understand it in the context of the original problem. Think of it as translating our mathematical result back into the language of the real world.

The Number Line: Visualizing the Solution

One of the best ways to understand our solution is to visualize it on a number line. A number line is a simple yet powerful tool that represents all real numbers as points on a line. Zero sits in the middle, positive numbers stretch out to the right, and negative numbers extend to the left. Our solution n ≤ -3 tells us that we're interested in all numbers that are less than or equal to -3. On the number line, this is represented by a closed circle (or a filled-in dot) at -3, indicating that -3 itself is included in the solution, and an arrow extending to the left, showing that all numbers smaller than -3 are also part of the solution.

Imagine painting the number line with a special color to mark our solution. We'd start by coloring the point at -3, and then we'd continue coloring everything to the left of -3. This colored portion represents all the numbers that satisfy our original condition.

Real-World Meaning: Numbers in Action

So, what does this mean in the grand scheme of things? Well, it means that if we pick any number from our solution set (like -3, -4, -5, -10, or even -100), and plug it back into our original statement, "Three times two less than a number is greater than or equal to five times the number," the statement will be true. Let's try it out with a number from our solution set, say -4:

• Our original statement translates to 3(n - 2) ≥ 5n.
• Substituting n = -4, we get 3(-4 - 2) ≥ 5(-4).
• Simplifying, we get 3(-6) ≥ -20, which is -18 ≥ -20.
• Is -18 greater than or equal to -20? Yes, it is! So, -4 is indeed a solution.

If we tried a number not in our solution set, like 0, the statement would not be true:

• Substituting n = 0, we get 3(0 - 2) ≥ 5(0).
• Simplifying, we get 3(-2) ≥ 0, which is -6 ≥ 0.
• Is -6 greater than or equal to 0? No, it's not! So, 0 is not a solution.

This confirms that our solution n ≤ -3 is the correct set of numbers that make our original statement true.

Conclusion: The Power of Mathematical Decoding

Guys, we've done it! We've successfully decoded the number puzzle, translated words into math, solved the inequality, and interpreted the solution. We've seen how mathematical statements can be broken down into smaller, manageable parts, and how the tools of algebra can help us find the hidden answers. More importantly, we've learned that math isn't just about numbers and symbols; it's about logical thinking, problem-solving, and understanding the world around us.

Remember, every mathematical problem is like a puzzle waiting to be solved. With a little bit of patience, the right tools, and a dash of curiosity, you can unlock any mathematical mystery. Keep practicing, keep exploring, and keep enjoying the power of mathematical decoding! You rock!