Sure! Let's walk through the process of converting the given inequality into interval notation and then discuss how we can graph it.
### Step-by-Step Solution
#### Step 1: Express the Inequality in Interval Notation
The inequality given is:
[tex]\[ -1 \leq x < 5 \][/tex]
This can be interpreted as [tex]\( x \)[/tex] being between [tex]\(-1\)[/tex] and [tex]\(5\)[/tex], inclusive of [tex]\(-1\)[/tex] but exclusive of [tex]\(5\)[/tex]. To express this in interval notation:
- The square bracket [tex]\([ \)[/tex] indicates that the endpoint is included (closed interval).
- The parenthesis [tex]\() \)[/tex] indicates that the endpoint is excluded (open interval).
So, the interval notation for the inequality [tex]\(-1 \leq x < 5\)[/tex] is:
[tex]\[ [-1, 5) \][/tex]
#### Step 2: Graph the Interval
1. Draw a Number Line:
- Draw a horizontal line and mark numbers on it, including [tex]\(-1\)[/tex] and [tex]\(5\)[/tex].
2. Mark the Endpoints:
- Place a solid dot at [tex]\(-1\)[/tex] to indicate that it is included in the interval.
- Place an open circle at [tex]\(5\)[/tex] to indicate that it is excluded from the interval.
3. Shade the Region Between the Endpoints:
- Shade the number line between [tex]\(-1\)[/tex] and [tex]\(5\)[/tex], showing all the numbers in this interval are part of the solution.
#### Example Graph:
```plaintext
<---------|---------|---------|---------|---------|---------|---------|--------->
-3 -2 -1 0 1 2 3 4 5
[--------------------------)
```
In the diagram:
- The solid dot at [tex]\(-1\)[/tex] means [tex]\(-1\)[/tex] is included.
- The open circle at [tex]\(5\)[/tex] means [tex]\(5\)[/tex] is excluded.
- The shaded region between [tex]\(-1\)[/tex] and [tex]\(5\)[/tex] represents all numbers [tex]\(x\)[/tex] that satisfy the inequality.
### Recap of Interval Notation and Graphing
- Interval Notation: [tex]\([-1, 5)\)[/tex]
- Graph Representation: Solid dot at [tex]\(-1\)[/tex], open circle at [tex]\(5\)[/tex], and shading between [tex]\(-1\)[/tex] and [tex]\(5\)[/tex].
This completes the detailed explanation of converting the inequality [tex]\(-1 \leq x < 5\)[/tex] into interval notation and graphing it.
