Solving Inequalities: Is W ≤ 8?

Hey guys! Let's dive into the world of inequalities and figure out if certain values of 'w' make the statement w ≤ 8 true. It's like a fun little puzzle where we check if a number is less than or equal to 8. This is a fundamental concept in mathematics, and understanding it will help you with more complex problems down the road. We'll be looking at specific values of 'w' and determining whether they fit the criteria. The process is pretty straightforward, but it's super important for building a solid foundation in algebra. Think of it as a stepping stone to bigger and better mathematical adventures! So, grab your pencils, and let's get started. We're going to break down each value and see if it plays nicely with the inequality w ≤ 8. Are you ready? Let's go!

Understanding the Inequality w ≤ 8

Alright, before we jump into the numbers, let's make sure we're all on the same page about what w ≤ 8 actually means. This inequality is saying that 'w' can be any number that is either less than 8 or equal to 8. The '≤' symbol is the key here; it tells us the relationship between 'w' and the number 8. Any number that fits this description is considered a solution to the inequality. For example, 7 is a solution because it's less than 8. 8 itself is also a solution because it's equal to 8. But 9 wouldn't be a solution because it's greater than 8. It's like a gatekeeper: only numbers that are on or below the gate are allowed in! It is an essential skill to identify solutions to inequalities. The concept of inequalities is very important in the field of mathematics, so get used to it! Understanding this is fundamental to grasping more complex mathematical concepts and problem-solving strategies, which is why we're going through this exercise. Let's make sure we've understood that any number lower than or equal to eight will satisfy the inequality, and then we are good to go.

Breaking Down the Symbol

The symbol '≤' is a combination of two ideas: '<' (less than) and '=' (equal to). When we see this symbol, we need to consider both possibilities. It's like having two criteria: is 'w' smaller than 8? Or, is 'w' exactly 8? If the answer to either of these questions is yes, then 'w' is a solution. This dual nature is crucial to remember. If the value of w is less than or equal to 8, then it satisfies the inequality. We're essentially looking for numbers that fall within a certain range. Remember that the equal part is also important. So, 8 itself is a valid solution. In other words, you have to remember that 8 is included. When we represent the solution of the inequality on a number line, we use a closed circle. It shows that the number is also included in the solution set. It's super easy, and we'll practice with some specific numbers in a bit. Just keep in mind that the symbol contains two possibilities. And, you'll be a pro in no time.

Testing Values of w

Now, let's put our knowledge to the test! We're going to examine specific values for 'w' and decide whether they satisfy the inequality w ≤ 8. Remember our rules: if 'w' is less than 8, or if 'w' is equal to 8, then it's a solution. We'll go through each value one by one and make our determination. Think of it as a series of mini-challenges. This exercise is important to learn how to deal with similar problems when the problem gets more complex. Are you ready to dive into the first number? Awesome, let's go!

Case 1: w = 6

Okay, our first value to check is w = 6. The question we need to ask is: Is 6 less than or equal to 8? Well, 6 is definitely less than 8. Since 6 is less than 8, it fits the description in the inequality. Therefore, 6 is a solution to w ≤ 8. Great job! You've successfully checked your first number. So, any number less than 8 will be a valid solution, for instance, 1, 2, 3, 4, 5, 6, 7. Remember that all of them, along with 8, are valid solutions. So, every number in the left direction from 8 on the number line will be valid. Keep up the good work; you're doing amazing.

Case 2: w = 12

Next up, we have w = 12. Now, is 12 less than or equal to 8? Nope! 12 is greater than 8. Because 12 is not less than or equal to 8, it doesn't meet our criteria. Therefore, 12 is not a solution to w ≤ 8. You can think of it as 12 being on the wrong side of the gate. This case shows us the boundaries of our solution set. So, any number greater than 8 will not be a solution. This is essential to understand, as it helps in identifying the range of solutions, especially in more complex problems. You got it; let's go on to our next number.

Case 3: w = 8

Finally, we have w = 8. Let's ask the question: is 8 less than or equal to 8? Well, 8 is equal to 8. Remember, in our inequality, we said that 'w' could be either less than or equal to 8. Since 8 satisfies the 'equal to' part of the inequality, it is a solution. This is a common point of confusion, so it's excellent that we're explicitly addressing it. Remember that the equal part is also important. So, 8 itself is a valid solution. It's like the gate being wide open for the number 8. Make sure you fully understand this, as it is the crux of the problem. This reinforces the importance of the 'equal to' part of the '≤' symbol. In summary, any number smaller than or equal to 8 is a valid solution.

Summary

So, to recap, let's go over what we learned: We investigated the inequality w ≤ 8 and tested three different values for 'w': 6, 12, and 8. We discovered that 6 is a solution because it's less than 8. 12 is not a solution because it's greater than 8. And 8 is a solution because it's equal to 8. You guys are awesome. We've taken a good step in understanding inequalities. Understanding these concepts forms the foundation for more advanced math topics. This understanding will be crucial as you continue your mathematical journey. Remember the basics, and you'll be golden. Keep practicing, and you'll become a pro at solving inequalities! Remember to always consider both the 'less than' and 'equal to' parts of the '≤' symbol. Keep up the fantastic work, and you'll be well on your way to mastering algebra. Keep practicing, and you'll become a pro at solving inequalities. And, keep having fun! Remember that the more you practice, the easier these problems will become.