Out of all the patients in a hospital 89 % are found to be...

Let $H$ and $L$ is set of people suffered from heart ailment and lungs infections respectively.

$n\left(H\right) = 89$

$n\left(L\right) = 98$

$n\left(H \cap L\right) = K$

$max\left\{n ( H ) , n\left(L\right)\right\} \le n\left(H \cup L\right) \le n\left(U\right)$

Where, $U$ is the universal set.

$⇒ 98 \le n\left(H \cup L\right) \le 100$

$⇒ 98 \le n\left(H\right) + n\left(L\right) - n\left(H \cap L\right) \le 100$

$⇒ 98 \le 89 + 98 - k \le 100$

$⇒ 0 \le 89 - k \le 2$

$⇒ - 89 \le - k \le - 87$

$⇒ 87 \le K \le 89$