Final answer:
To insert six rational numbers between -1/2 and -2/3, we make the denominators the same to create a sequence with a common difference, resulting in the rational numbers -11/6, -10/6, -9/6, -8/6, -7/6, and -5/6.
Step-by-step explanation:
To insert six rational numbers between -1/2 and -2/3, we need to find a common difference that we can use to define these numbers. First, we convert these fractions to have the same denominator. We can do this by multiplying the denominators together, which gives us a common denominator of 6. So, our two fractions become -3/6 and -4/6. We can then create six fractions with this common denominator that fall between the two. The numerators of these new fractions will be -11/6, -10/6, -9/6, -8/6, -7/6, and -5/6.It's important to ensure that these fractions are indeed between the initial two fractions, and we can do this by checking that their numerical values fall between -0.5 and -0.666.... By observing the sequence, we confirm that each subsequent fraction is 1/6 greater than the last, and all are between the limits of -1/2 and -2/3.