Hint: In this question we have to find five rational numbers between the given numbers. A rational number is one which is of the form $\dfrac{p}{q}$ where $q \ne 0$, use this basic definition of rational numbers. Firstly figure out all the decimal integers (only 5 required) then convert them into fractional form to get the answers.
Complete step-by-step answer: As we know rational numbers are those which are written in the form of $\dfrac{p}{q}$, where $q \ne 0$ and $\dfrac{p}{q}$ is written in lowest form, such that $p$ and $q$ have not no common factors except 1.
For example $\dfrac{2}{3}$ as this fraction is written in lowest form and does not have any common factors except 1, so this is a rational number.
Now we have to find out the rational numbers between -2 and -1 i.e. greater than -2 and less than -1.
So consider any five numbers between -2 and -1 which is $\left\{ { - 1.1, - 1.2, - 1.3, - 1.4, - 1.5} \right\}$
So, \[\left\{ { - \dfrac{3}{2}, - \dfrac{7}{5}, - \dfrac{{13}}{{10}}, - \dfrac{6}{5}, - \dfrac{{11}}{{10}}} \right\}\] are the required five rational numbers between (-2 and -1).
Note: Whenever we face such types of problems the key concept is simply to have the gist of the basic understanding of rational numbers. There can be many decimal integers between a given interval but however be concerned only about the number of rational numbers being asked, this will help you getting on the right track to reach the answer.