Video Transcript
Which of the following is
true? Is it (A) five is an element of
the open interval from root six to root 26? Or is it (B) five is not an
element of the open interval from root six to root 26?
When we read these two options,
we read them, respectively, as five is an element and five is not an element of
that interval. In other words, since the
interval is open, in option (A), we’re saying five is between root six and root
26. And in option (B), we’re saying
it’s not between these two numbers. In order to identify which is
true, we could use a calculator to evaluate root six and root 26.
We’re going to use a
non-calculator method though. This method involves quoting
the first six square numbers. One squared is one, two squared
is four, all the way up to six squared is 36. Then, we compare these to the
first number in each interval. Since taking the square root is
the inverse operation to squaring a number, we can say that the square root of
six must be between the square root of four and the square root of nine. In other words, the square root
of six is between two and three. Then, we look at the second
number in our interval, the square root of 26. The square root of 26 must be
between the square root of 25 and the square root of 36. In other words, the square root
of 26 is greater than five and less than six.
Now, we have indeed shown that
five is less than the square root of 26. And of course, since five is
bigger than three, by definition, five must also be bigger than the square root
of six. If we represent this in
inequality notation, we can say that five is greater than root six and less than
root 26 and that in turn means we can use interval notation to say that five is
an element of the open interval from root six to root 26. The correct answer is (A).
