Finding X Values For Function Y = 2(2x - 7) Non-Negative And Less Than 4

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Hey guys! Let's dive into a fun math problem today. We're going to figure out for what values of x the function y = 2(2x - 7) behaves in certain ways. Specifically, we want to know when it gives us non-negative values (meaning zero or positive) and when its values are no bigger than 4. Sounds interesting, right? Let's break it down step by step.

Understanding the Function

First things first, let's get comfy with our function: y = 2(2x - 7). This is a linear function, which means it's going to give us a straight line when we graph it. Linear functions are super cool because they have a constant rate of change. In our case, for every increase in x, y changes at a predictable rate. Now, to tackle our problem, we'll need to use some inequalities. Inequalities are mathematical expressions that use symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). They're perfect for describing situations where we have a range of values rather than a single, exact value. So, with our function in mind, let’s see how inequalities can help us find the values of x that make y non-negative.

Solving for Non-Negative Values (y ≥ 0)

Okay, so the first part of our mission is to find the values of x that make y non-negative. Non-negative means y is either zero or a positive number. Mathematically, we write this as y ≥ 0. Let's plug our function into this inequality: 2(2x - 7) ≥ 0. Now, we need to solve for x. The first step is to get rid of that 2 outside the parentheses. We can do this by dividing both sides of the inequality by 2. Remember, when we divide or multiply both sides of an inequality by a positive number, the direction of the inequality stays the same. So, we get: 2x - 7 ≥ 0. Awesome! We're getting closer. Now, let's isolate x. We can do this by adding 7 to both sides: 2x ≥ 7. Finally, to get x by itself, we divide both sides by 2: x ≥ 7/2. Or, if you prefer decimals, x ≥ 3.5. What does this mean? It means that for any value of x that is greater than or equal to 3.5, our function y will give us a non-negative value. Cool, right? You can even try plugging in a few values to see for yourself. For example, if x = 4, then y = 2(2*4 - 7) = 2, which is positive. If x = 3.5, then y = 2(2*3.5 - 7) = 0. Perfect!

Finding Values Not Greater Than 4 (y ≤ 4)

Alright, let's move on to the second part of our quest: finding the values of x that make y not greater than 4. This means y can be 4 or any number smaller than 4. In math terms, we write this as y ≤ 4. Just like before, let's substitute our function into this inequality: 2(2x - 7) ≤ 4. Now, let's solve for x. Again, the first step is to divide both sides by 2: 2x - 7 ≤ 2. Great! Now, we'll add 7 to both sides to isolate the term with x: 2x ≤ 9. Finally, divide both sides by 2 to get x by itself: x ≤ 9/2. As a decimal, this is x ≤ 4.5. So, what does this tell us? It means that for any value of x that is less than or equal to 4.5, our function y will give us a value that is not greater than 4. Let's test this out with a couple of examples. If x = 4, then y = 2(2*4 - 7) = 2, which is less than 4. If x = 4.5, then y = 2(2*4.5 - 7) = 4. Awesome, it works!

Putting It All Together

So, we've successfully tackled both parts of our problem! We've found that:

  • For y to be non-negative (y ≥ 0), x must be greater than or equal to 3.5 (x ≥ 3.5).
  • For y to be not greater than 4 (y ≤ 4), x must be less than or equal to 4.5 (x ≤ 4.5).

These ranges of x values are where our function behaves in the ways we described. This is a fantastic example of how we can use inequalities to understand the behavior of functions. Remember, functions are the heart of mathematics, and understanding them helps us solve all sorts of real-world problems.

Why is This Important?

You might be wondering, "Okay, this is cool, but why does it matter?" Well, understanding when a function is positive, negative, or within certain bounds is super important in many fields. For example:

  • In physics, you might use this to determine when an object's velocity is positive (moving forward) or negative (moving backward).
  • In economics, you could use it to find the range of prices that result in a profit for a business.
  • In engineering, you might need to ensure that a certain value stays within safe operating limits.

So, the skills we've used today are actually quite powerful and have lots of applications. Plus, they help build your problem-solving muscles, which is always a good thing!

Tips for Solving Similar Problems

Now that we've worked through this problem together, let's talk about some tips that can help you tackle similar questions in the future. These steps are like a roadmap for solving inequality problems, so keep them handy:

  1. Understand the Question: Make sure you know exactly what the question is asking. Are you looking for when a function is positive, negative, greater than a certain value, or something else? Identify the key conditions.
  2. Set Up the Inequality: Translate the conditions into a mathematical inequality. This is where you'll use symbols like ≥, ≤, >, or <. Be sure to use the right symbol to accurately represent the problem.
  3. Solve for the Variable: Use algebraic techniques to isolate the variable you're interested in (in our case, x). Remember to perform the same operations on both sides of the inequality to keep it balanced. A key thing to remember is that if you multiply or divide both sides by a negative number, you need to flip the direction of the inequality sign.
  4. Interpret the Solution: Once you've solved for the variable, think about what the solution means in the context of the problem. What range of values satisfies the conditions? Write your answer clearly and concisely.
  5. Check Your Work: It's always a good idea to check your solution by plugging in a few values from the range you found. This helps you make sure your answer makes sense and that you haven't made any mistakes.

By following these steps, you'll be well-equipped to solve all sorts of inequality problems. Remember, practice makes perfect, so don't be afraid to try lots of examples!

Wrapping Up

So, there you have it! We've successfully navigated the world of functions and inequalities to find the values of x that make y = 2(2x - 7) behave in specific ways. We've seen how to determine when y is non-negative and when it's not greater than 4. These are important skills that can be applied in many different areas of math and beyond. I hope this has been a helpful and fun journey for you guys. Remember, math is all about understanding patterns and relationships, and with a little practice, you can become a math whiz in no time! Keep exploring, keep questioning, and most importantly, keep learning! You've got this! If you have any questions or want to explore more math problems, feel free to ask. Happy problem-solving!