To solve the inequality [tex]\( x + \frac{1}{2} \leq -3 \)[/tex] or [tex]\( x - 3 > -2 \)[/tex], we need to solve each part of the inequality separately.
### Solving the first inequality:
[tex]\[ x + \frac{1}{2} \leq -3 \][/tex]
1. Subtract [tex]\(\frac{1}{2}\)[/tex] from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x \leq -3 - \frac{1}{2}
\][/tex]
2. Simplify the right-hand side:
[tex]\[
x \leq -3.5
\][/tex]
### Solving the second inequality:
[tex]\[ x - 3 > -2 \][/tex]
1. Add 3 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x > -2 + 3
\][/tex]
2. Simplify the right-hand side:
[tex]\[
x > 1
\][/tex]
### Combining the results:
The solutions to the inequalities are:
[tex]\[ x \leq -3.5 \text{ or } x > 1 \][/tex]
So, the final solution to the inequality is:
[tex]\[ x \leq -3.5 \text{ or } x > 1 \][/tex]
