Range Problem: Finding Max/Min Values Of X
Let's break down this math problem step-by-step to figure out how to find the maximum and minimum possible values of 'x' when we know the range of a set of numbers. So, guys, grab your thinking caps, and let's dive in!
Understanding the Range
First, it's super important to understand what "range" means in math. The range of a set of numbers is simply the difference between the largest and smallest numbers in that set. It tells us how spread out the numbers are. For example, if you have the numbers 3, 5, 8, and 10, the range is 10 - 3 = 7. Easy peasy, right?
In our problem, we're told that the range of the numbers X, 12, 9, 10, 15, and 13 is 8. This means that the difference between the biggest and smallest number in this list is 8. But here's the catch: we don't know what X is! That's what we need to figure out. To solve for the maximum and minimum possible values of 'x' when we know the range of a set of numbers. First, it's super important to understand what "range" means in math. The range of a set of numbers is simply the difference between the largest and smallest numbers in that set. It tells us how spread out the numbers are. For example, if you have the numbers 3, 5, 8, and 10, the range is 10 - 3 = 7. Easy peasy, right?
Identifying the Known Numbers
Let's take a closer look at the numbers we do know: 12, 9, 10, 15, and 13. We need to find the largest and smallest numbers among these. By inspection:
- The largest number is 15.
- The smallest number is 9.
These values will be crucial when we consider the possible values of X. Determining these known values is a critical step to solve for the maximum and minimum possible values of 'x'.
Finding the Maximum Value of X
Now, let's think about how to find the maximum possible value of X. Remember, the range is the difference between the largest and smallest numbers. If X is the largest number, then the range would be X - 9 (since 9 is the smallest number we already know). We're told that the range is 8, so we can set up an equation:
- X - 9 = 8
To solve for X, we simply add 9 to both sides of the equation:
- X = 8 + 9
- X = 17
So, if X is the largest number in the set, its maximum possible value is 17. This makes sense because if we put 17 into the set, the numbers would be 17, 12, 9, 10, 15, and 13. The range would indeed be 17 - 9 = 8. To reiterate, we solve for the maximum value of 'x' by using the equation X - 9 = 8, resulting in X = 17. Understanding the maximum value helps us define the upper limit for 'x' within the given range constraint.
Finding the Minimum Value of X
Okay, now let's figure out the minimum possible value of X. In this case, X would be the smallest number in the set. If X is the smallest number, then the range would be 15 - X (since 15 is the largest number we already know). Again, we know the range is 8, so we can set up another equation:
- 15 - X = 8
To solve for X, we can first subtract 15 from both sides:
- -X = 8 - 15
- -X = -7
Then, multiply both sides by -1 to get X:
- X = 7
So, if X is the smallest number in the set, its minimum possible value is 7. Let's check if this makes sense. If we put 7 into the set, the numbers would be 7, 12, 9, 10, 15, and 13. The range would be 15 - 7 = 8. Spot on! This part involves finding the minimum possible value for 'x'. We used the equation 15 - X = 8 to find that X = 7. This means that if 'x' is the smallest number in the set, its value must be 7 to maintain the range of 8.
Considering X Within the Existing Range
But hold on a second! Before we declare victory, we need to consider something important. What if X falls within the range of the numbers we already have? Specifically, what if including X doesn't actually change the largest or smallest number in the set? In other words, what if X is between 9 and 15?
Let's think about the implications:
- If X is greater than 9 but less than 15: The smallest number is still 9, and the largest number is still 15. This means the range is 15 - 9 = 6. But we know the range must be 8. So, X cannot fall strictly between 9 and 15. This constraint is vital because it narrows down the potential values of 'x'. By recognizing that 'x' cannot fall strictly between 9 and 15, we've eliminated a range of possible values, making it easier to pinpoint the exact maximum and minimum.
This consideration reinforces that X must either be the smallest or largest number in the set to maintain the given range of 8. To put it another way, the value of X must affect the span of the data to align with the stated range.
Conclusion
Alright, folks, we've cracked the case! To recap:
- The maximum possible value of X is 17.
- The minimum possible value of X is 7.
Therefore, X can either be 17 (making it the largest number) or 7 (making it the smallest number) to satisfy the condition that the range of the set is 8. Hopefully, this detailed explanation helped you understand how to solve range problems. Understanding how each number affects the set and overall range is key. Keep practicing, and you will become a math whiz in no time! By combining a clear understanding of what range means with careful algebraic manipulation, we can confidently find the possible values of an unknown within a data set. To summarize, the maximum possible value of X is 17, and the minimum possible value of X is 7, ensuring that the range of the set remains 8.