Find the value of 𝑥 given that the range of the values seven, nine, five, four, six, two, and 𝑥 is eight and 𝑥 is greater than two.
In this problem, we have a data set that consists of seven different numbers, six of these numbers we know their value and one that we don’t, which is 𝑥. Knowing that the range is eight will allow us to work out this unknown value. Let’s begin by putting the six values that we know in order from smallest to largest. Usually when we’re working with the range, we don’t need to put the values in order. We just need to know the highest and the lowest values. This is because the range is calculated by the highest value subtract the lowest value.
So, when it comes to working out the value of 𝑥, there are three different scenarios. The first scenario is if we have these numbers in order that the value of 𝑥 is smaller than the value of two. For example, 𝑥 could be negative 10, zero, or even one. However, we are told that 𝑥 must be greater than two, and so this situation cannot occur.
The second scenario is that 𝑥 takes any value from two up to nine. For example, 𝑥 could appear in the list as two, four, 𝑥, five, six, seven, and nine. And what would the range be in this case? Well, the range is the highest value, which would be nine, subtract the lowest value, which is two, which would give us a value of seven. However, we were told that the range of these values is eight. And so, that means that 𝑥 cannot be some value between two and nine. Even if 𝑥 was either of the values of nine or two, then the range here would still be seven and not the value of eight.
And so, this brings us to the third scenario, which is that the value of 𝑥 is larger than the value nine. To work out the range in this situation, we would have the largest value, which we’re saying would be 𝑥, subtract the smallest value, which would be two. Now, we can use the given information that the range is equal to eight. So, we would have the equation that eight is equal to 𝑥 minus two. Adding two to both sides then gives us that 10 is equal to 𝑥 or 𝑥 is equal to 10.
And so, this unknown data value of 𝑥 must be equal to 10, since this is the only way that this data set will have a range of eight and a value of 𝑥 greater than two.