Answer
To represent the inequalities on a number line:
1. **For \( x > 5 \):**
- Draw an open circle at 5.
- Shade the line to the right of 5.
2. **For \( y \geq -4 \):**
- Draw a closed circle at -4.
- Shade the line to the right of -4.
These representations show all values greater than 5 for \( x \) and all values greater than or equal to -4 for \( y \).
Solution
To represent the inequalities on a number line, we will analyze each inequality step by step.
### Inequality 4: \( x > 5 > -1 \)
This inequality can be broken down into two parts:
1. \( x > 5 \)
2. \( 5 > -1 \) (which is always true)
The relevant part for the number line is \( x > 5 \). This means that \( x \) can take any value greater than 5.
**Representation on the number line:**
- We will draw an open circle at 5 (indicating that 5 is not included).
- We will shade the line to the right of 5 to indicate all values greater than 5.
### Inequality 6: \( y \geq -4 \)
This inequality states that \( y \) is greater than or equal to -4.
**Representation on the number line:**
- We will draw a closed circle at -4 (indicating that -4 is included).
- We will shade the line to the right of -4 to indicate all values greater than or equal to -4.
### Summary of Representations:
1. For \( x > 5 \):
- Open circle at 5, shading to the right.
2. For \( y \geq -4 \):
- Closed circle at -4, shading to the right.
These representations can be visualized on a number line as follows:
```
Number Line for x > 5:
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-2 -1 0 1 2 3 4 5 6 7 8 9 10
(--------------------->
Number Line for y ≥ -4:
<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4
●--------------------->
```
In the above representation:
- The open circle at 5 indicates that 5 is not included in the solution for \( x \).
- The closed circle at -4 indicates that -4 is included in the solution for \( y \).
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