Question

Write the following as intervals :
(i) \[\left\{ {{\text{x : x }} \in {\text{ R , 4 < x }} \leqslant {\text{ 6}}} \right\}\]
(ii) \[\left\{ {{\text{x : x }} \in {\text{ R , - 12 < x < - 10}}} \right\}\]
(iii) \[\left\{ {{\text{x : x }} \in {\text{ R , 0 }} \leqslant {\text{ x < 7}}} \right\}\]
(iv) \[\left\{ {{\text{x : x }} \in {\text{ R , 3 }} \leqslant {\text{ x }} \leqslant {\text{ 4}}} \right\}\]

Answer 1

Hint: So for converting in interval form we have to write it in the brackets as if use $\left( {} \right]$ this bracket mean that the digit in the LHS is not included in the interval while the digit in RHS is included in the interval , Similarly for this $\left[ {} \right)$ , In this $\left[ {} \right]$ bracket both side is included while in $\left( {} \right)$ in this both digit not included .

Complete step-by-step answer:
As we have to convert these in the interval form ,
In the part (i) \[\left\{ {{\text{x : x }} \in {\text{ R , 4 < x }} \leqslant {\text{ 6}}} \right\}\]
the interval form of this will be , $\left( {4,6} \right]$
we use $\left( {} \right]$ this kind of bracket because $4$ is not included after $4$ all the real numbers are included till $6$ in the interval and $6$ will be included .
In the part (ii) \[\left\{ {{\text{x : x }} \in {\text{ R , - 12 < x < - 10}}} \right\}\]
In the interval we write it as $\left( { - 12, - 10} \right)$
we use $\left[ {} \right)$ this bracket $ - 12$ and $ - 10$ both are not included but in between $ - 12$ and $ - 10$ all the real numbers are included in the interval .
In the part (iii) \[\left\{ {{\text{x : x }} \in {\text{ R , 0 }} \leqslant {\text{ x < 7}}} \right\}\]
In the interval form we write it as $\left[ {0,7} \right)$
As $0$ is included while $7$ is not .
In the part (iv) \[\left\{ {{\text{x : x }} \in {\text{ R , 3 }} \leqslant {\text{ x }} \leqslant {\text{ 4}}} \right\}\]
In the interval form if we write it $\left[ {3,4} \right]$
As both $3,4$ are included and in between all the real numbers are included .

Note: If we have to write the real numbers in the interval form we write it as $\left( { - \infty ,\infty } \right)$ , As always remember that with $\infty $ we did not use the closed bracket that is $\left[ {} \right.$ or $\left. {} \right]$ because infinity is not known quantity so we always use open bracket means $\left( {} \right)$