Given : P = `{x : 5 lt 2x - 1 le 11, x in R}`
`Q = {x : 1 le 3 + 4x lt 23, x in I}`
where R = (real numbers) and I = (integers).
Represent P and Q on two different number lines. Write down the elements of `P cap Q`.

Answer

To solve the problem, we need to find the sets \( P \) and \( Q \) based on the given inequalities, represent them on number lines, and then find the intersection of these two sets.

### Step 1: Solve for Set \( P \)

Given the inequality for \( P \):
\[ 5 < 2x - 1 \leq 11 \]

**Step 1.1:** Break it into two parts:
1. \( 5 < 2x - 1 \)
2. \( 2x - 1 \leq 11 \)

**Step 1.2:** Solve the first part:
\[ 5 < 2x - 1 \implies 5 + 1 < 2x \implies 6 < 2x \implies x > 3 \]

**Step 1.3:** Solve the second part:
\[ 2x - 1 \leq 11 \implies 2x \leq 11 + 1 \implies 2x \leq 12 \implies x \leq 6 \]

**Step 1.4:** Combine the results:
\[ 3 < x \leq 6 \]
Thus, the set \( P \) can be represented as:
\[ P = \{ x : 3 < x \leq 6, x \in \mathbb{R} \} \]

### Step 2: Solve for Set \( Q \)

Given the inequality for \( Q \):
\[ 1 \leq 3 + 4x < 23 \]

**Step 2.1:** Break it into two parts:
1. \( 1 \leq 3 + 4x \)
2. \( 3 + 4x < 23 \)

**Step 2.2:** Solve the first part:
\[ 1 \leq 3 + 4x \implies 1 - 3 \leq 4x \implies -2 \leq 4x \implies x \geq -\frac{1}{2} \]

**Step 2.3:** Solve the second part:
\[ 3 + 4x < 23 \implies 4x < 23 - 3 \implies 4x < 20 \implies x < 5 \]

**Step 2.4:** Combine the results:
\[ -\frac{1}{2} \leq x < 5 \]
Since \( x \) must be an integer, the set \( Q \) can be represented as:
\[ Q = \{ x : -1 \leq x < 5, x \in \mathbb{I} \} = \{-1, 0, 1, 2, 3, 4\} \]

### Step 3: Represent \( P \) and \( Q \) on Number Lines

**For Set \( P \):**
- The interval \( (3, 6] \) means we shade the region between 3 (not included) and 6 (included).

**For Set \( Q \):**
- The interval \( [-1, 5) \) means we shade the region between -1 (included) and 5 (not included).

### Step 4: Find \( P \cap Q \)

To find the intersection \( P \cap Q \), we look for common elements in both sets.

- \( P = (3, 6] \) includes \( 4, 5, 6 \)
- \( Q = \{-1, 0, 1, 2, 3, 4\} \)

The common element is:
\[ P \cap Q = \{4\} \]

### Final Answer

The elements of \( P \cap Q \) are:
\[ P \cap Q = \{4\} \]