on a number line show the intersection and the union of the following intervals. (-3, infinity) and (4, infinity)

### Intersection and Union of the Intervals (-3, ∞) and (4, ∞)

Let's first break down the intervals:

- **Interval 1**: (-3, ∞) includes all real numbers greater than -3, but **not including** -3 itself.
- **Interval 2**: (4, ∞) includes all real numbers greater than 4, but **not including** 4 itself.

#### 1. **Intersection**:  
The **intersection** of two sets is the set of elements that are common to both. In this case, we are looking for the numbers that are in both intervals.

- Both intervals overlap only for numbers greater than 4, because:
  - The first interval includes numbers greater than -3.
  - The second interval includes numbers greater than 4.

Therefore, the **intersection** is:
\[ (4, ∞) \]

#### 2. **Union**:  
The **union** of two sets is the set of all elements that are in either one or both sets. In this case, it includes all numbers from both intervals.

- The first interval starts at -3, and the second interval starts at 4.  
- Since the first interval already covers all numbers greater than -3 (which includes numbers from 4 onwards as well), the **union** will be the entire first interval.

Therefore, the **union** is:
\[ (-3, ∞) \]

### Number Line Representation:
- For the **intersection**: a line segment starting at 4 (but not including 4) and extending to infinity.
- For the **union**: a line segment starting at -3 (but not including -3) and extending to infinity.

Would you like a visual illustration for the number line?

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### Related Questions:
1. What is the intersection of (-2, ∞) and (1, ∞)?
2. What is the union of (-5, 2) and (0, 6)?
3. How do you find the intersection of two intervals that are disjoint?
4. What is the difference between an open and closed interval?
5. Can there be an interval with a finite endpoint in both directions?

**Tip:** When finding the union of intervals, always look at the earliest start and the latest endpoint of the combined intervals.

On a number line, show the intersection and the union of the following intervals: (-3, infinity) and (4, infinity).

Explore the intersection and union of the intervals (-3, ∞) and (4, ∞) with a clear explanation and number line representation. This problem covers essential set theory concepts and interval notation, suitable for students in Grades 8-10.

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