Solve the following inequality and show the solution on the number line - Junior Cycle Mathematics - Question a - 2013

We start with the inequality:

$-2 \frac{1}{2} < \frac{1}{2} x - 3 < 1$

1. Rewrite the first part of the inequality:

   $-2.5 < \frac{1}{2} x - 3$

   Adding 3 to all sides:

   $0.5 < \frac{1}{2} x$

   Multiplying by 2:

   $1 < x$

2. Now for the second part of the inequality:

   $\frac{1}{2} x - 3 < 1$

   Adding 3 to both sides:

   $\frac{1}{2} x < 4$

   Multiplying by 2:

   $x < 8$

3. Combining the two results gives:

   $1 < x < 8$

4. Thus, the integer solutions for $x \in \mathbb{N}$ are:

   $x = 2, 3, 4, 5, 6, 7$

5. On the number line, you highlight the integers between 1 and 8 (not including 1 and 8).