Given $P = \left\{x \; \Big| \; 9 < 2x -1 \leq 13, \; x \in R \right\}$, $\;$ $Q = \left\{x \; \Big| \; -5 \leq 3 + 4x < 15, \; x \in I \right\}$ $\;$ where $R$ is the set of real numbers and $I$ is the set of integers.
Represent $P$ and $Q$ on different number lines.
Write down the elements of $P \cap Q$.
$P = \left\{x \; \Big| \; 9 < 2x -1 \leq 13, \; x \in R \right\}$
Consider $\;$ $9 < 2x - 1$
i.e. $\;$ $10 < 2x$ $\implies$ $5 < x$ $\;\;\; \cdots \; (1)$
Consider $\;$ $2x - 1 \leq 13$
i.e. $\;$ $2x \leq 14$ $\implies$ $x \leq 7$ $\;\;\; \cdots \; (2)$
$\therefore \;$ We have from equations $(1)$ and $(2)$ the solution set of $P$ is
$\left\{x \; \Big | \; 5 < x \leq 7, \; x \in R\right\}$
i.e. $\;$ $-8 \leq 4x$ $\implies$ $-2 \leq x$ $\;\;\; \cdots \; (3)$
Consider $\;$ $3 + 4x < 15$
i.e. $\;$ $4x < 12$ $\implies$ $x < 3$ $\;\;\; \cdots \; (4)$
$\therefore \;$ We have from equations $(3)$ and $(4)$ the solution set of $P$ is
$\left\{x \; \Big | \; -2 \leq x < 3, \; x \in I \right\}$
i.e. $\;$ $\left\{-2, -1, 0, 1, 2 \right\}$
i.e. $\;$ $P \cap Q = \phi$ (null set)