\(P = {x : 1 \leq x \leq 6}\) and \(Q = {x : 2 < x < 9}\) where \(x \in R\), find \(P \cap Q\).

Answer Details

\(P = {x : 1 \leq x \leq 6}\) means that P is the set of all real numbers between 1 and 6 (including 1 and 6). Similarly, \(Q = {x : 2 < x < 9}\) means that Q is the set of all real numbers between 2 and 9 (excluding 2 and 9). To find \(P \cap Q\), we need to find the set of all elements that are common to both P and Q. From the definitions of P and Q, we can see that the range of values that satisfy both P and Q is from 2 to 6 (including 2 and 6). Therefore, \(P \cap Q\) is the set of all real numbers between 2 and 6 (including 2 and 6). Hence, the correct option is (d) \({x : 2 < x \leq 6}\).