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Question: Which of the following represents a rational number between \[ - 6\] and \[ - 7\] ? A. \[\dfrac{{ ...

Which of the following represents a rational number between −6 - 6 and −7 - 7 ?
A. −6−72\dfrac{{ - 6 - 7}}{2}
B. −6+72\dfrac{{ - 6 + 7}}{2}
C. 6+72\dfrac{{6 + 7}}{2}
D. −6−7 - 6 - 7

Explanation

Solution

In the above question, we are given two negative integers as −6 - 6 and −7 - 7 . We have to determine which of the following four rational numbers lie between these two numbers. The four options are given as −6−72\dfrac{{ - 6 - 7}}{2} , −6+72\dfrac{{ - 6 + 7}}{2} , 6+72\dfrac{{6 + 7}}{2} , −6−7 - 6 - 7 . In order to approach the solution, first we have to check the value of each rational number and then see if it lies between −6 - 6 and −7 - 7 or not.

Complete step by step answer:
Given two negative integers are −6 - 6 and −7 - 7. We have to determine if the given four options lie between these two integers or not. Since, both the integers are negative, therefore any real number lying between −6 - 6 and −7 - 7 is also obviously a negative real number. This information will help us easily identify some options which are a positive number.

Now, let us check the given options one by one whether they lie between −6 - 6 and −7 - 7 or not.
A. −6−72\dfrac{{ - 6 - 7}}{2}
Here, the numerator is equal to the sum of two numbers i.e. −6−7 - 6 - 7 and in the denominator there is 22 . That means the number −6−72\dfrac{{ - 6 - 7}}{2} is nothing else but actually the mean of −6 - 6 and −7 - 7 . Hence, it lies between −6 - 6 and −7 - 7. Also,
−6−72=−132=−6.5\dfrac{{ - 6 - 7}}{2} = - \dfrac{{13}}{2} = - 6.5

B. −6+72\dfrac{{ - 6 + 7}}{2}
Here the numerator is −6+7=1 - 6 + 7 = 1 . That means the given rational number is positive.Therefore it does not lie between −6 - 6 and −7 - 7 .

C. 6+72\dfrac{{6 + 7}}{2}
Also in this case, the numerator is positive again. Hence, the given rational number is also positive therefore it does not lie between −6 - 6 and −7 - 7 .

D. −6−7 - 6 - 7
Here the sum of −6 - 6 and −7 - 7 is given only. That is obviously less than both −6 - 6 and −7 - 7 as −6−7=−13 - 6 - 7 = - 13 . Therefore, it does not lie between −6 - 6 and −7 - 7. Hence, only the first rational number, that is −6−72\dfrac{{ - 6 - 7}}{2} , lies between the given two negative integers −6 - 6 and −7 - 7 .

Therefore, only the first option A is correct.

Note: There are in fact, infinitely many rational numbers lying between any two distinct integers. And also, similarly there are infinitely many rational or irrational numbers lying between any two distinct rational or irrational numbers. Overall in the end we can say that there are infinitely many real numbers between any two distinct real numbers.