Question: a) Solve 6 < \frac{x+5}{2} \leq 8
b) Draw your answer to part a) on a number line.
1. **State the problem:**
Solve the compound inequality $$6 < \frac{x+5}{2} \leq 8$$ and then represent the solution on a number line.
2. **Recall the rule:**
For inequalities involving fractions, multiply all parts by the denominator to eliminate the fraction, being careful to maintain inequality directions.
3. **Multiply all parts by 2:**
$$6 \times 2 < \frac{x+5}{2} \times 2 \leq 8 \times 2$$
$$12 < x + 5 \leq 16$$
4. **Isolate $x$ by subtracting 5 from all parts:**
$$12 - 5 < x + 5 - 5 \leq 16 - 5$$
$$7 < x \leq 11$$
5. **Interpret the solution:**
$x$ is greater than 7 but less than or equal to 11.
6. **Graphical representation:**
On the number line, this corresponds to an open circle at 7 (since $x$ is strictly greater than 7) and a closed circle at 11 (since $x$ can be equal to 11), with the segment between them shaded.
**Final answer:**
$$7 < x \leq 11$$