To solve the compound inequality [tex]\( -3 + n < -4 \text{ or } n + 3 \geq 4 \)[/tex], we need to solve each part separately and then find the union of the solutions. Let's take it step by step.
### Part 1: Solve [tex]\( -3 + n < -4 \)[/tex]
1. Start with the inequality:
[tex]\[
-3 + n < -4
\][/tex]
2. To isolate [tex]\( n \)[/tex], add 3 to both sides:
[tex]\[
-3 + n + 3 < -4 + 3
\][/tex]
3. Simplify the expression:
[tex]\[
n < -1
\][/tex]
The solution for the first part is [tex]\( n < -1 \)[/tex].
### Part 2: Solve [tex]\( n + 3 \geq 4 \)[/tex]
1. Start with the inequality:
[tex]\[
n + 3 \geq 4
\][/tex]
2. To isolate [tex]\( n \)[/tex], subtract 3 from both sides:
[tex]\[
n + 3 - 3 \geq 4 - 3
\][/tex]
3. Simplify the expression:
[tex]\[
n \geq 1
\][/tex]
The solution for the second part is [tex]\( n \geq 1 \)[/tex].
### Combine the Solutions
Since the original compound inequality is connected by an "or", we combine the solutions from both parts. The complete solution is:
[tex]\[
n < -1 \quad \text{or} \quad n \geq 1
\][/tex]
### Graphing the Solution
To graph this solution on a number line:
1. For [tex]\( n < -1 \)[/tex]:
- Draw a circle at [tex]\( n = -1 \)[/tex].
- Shade to the left of [tex]\( -1 \)[/tex] (indicating all values less than [tex]\(-1\)[/tex]).
2. For [tex]\( n \geq 1 \)[/tex]:
- Draw a solid circle (or a filled-in circle) at [tex]\( n = 1 \)[/tex].
- Shade to the right of [tex]\( 1 \)[/tex] (indicating all values greater than or equal to [tex]\( 1 \)[/tex]).
Here's a visual representation of the solution on a number line:
```
<=====(----+----)============>
-1 0 1
```
- The left arrow indicates all numbers less than [tex]\(-1\)[/tex].
- The filled circle at [tex]\(1\)[/tex] indicates [tex]\(n = 1\)[/tex] is included.
- The shading to the right of [tex]\(1\)[/tex] indicates all numbers greater than or equal to [tex]\(1\)[/tex].
