Express the interval using two different representations.

x, is less than or equal to, minus, 2, or , x, is greater than or equal to, 0

*x*≤−2 or *x*≥0

Answer

Attempt 1 out of 2

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Interval Notation:       

Number Line:

To express the interval \(x \leq -2\) or \(x \geq 0\) using two different representations, we will use **interval notation** and a **number line**.

### 1. Interval Notation

The given conditions are:
1. \(x \leq -2\)
2. \(x \geq 0\)

In interval notation:

- \(x \leq -2\) is written as \((- \infty, -2]\). This includes all values from \(-\infty\) to \(-2\), including \(-2\) itself.
- \(x \geq 0\) is written as \([0, \infty)\). This includes all values from \(0\) to \(\infty\), including \(0\) itself.

Since these two intervals do not overlap and represent an "or" condition, we combine them using the union symbol \( \cup \). Thus, the interval notation is:

\[
(-\infty, -2] \cup [0, \infty)
\]

### 2. Number Line Representation

On a number line:

- We have a closed circle at \(x = -2\) and shade all values to the left (towards negative infinity) to represent \(x \leq -2\).
- Similarly, we have a closed circle at \(x = 0\) and shade all values to the right (towards positive infinity) to represent \(x \geq 0\).

The number line would look like this:

```
<====●------------------●====>
     -2                  0
```

Where:
- The left part from \(-2\) extending to \( -\infty \) is shaded with a closed circle at \(-2\) to indicate that \(-2\) is included.
- The right part from \(0\) extending to \( \infty \) is also shaded with a closed circle at \(0\) to indicate that \(0\) is included.

### Summary
- **Interval Notation:** \((- \infty, -2] \cup [0, \infty)\)
- **Number Line:** Shaded regions to the left of \(-2\) and to the right of \(0\) with closed circles at \(-2\) and \(0\).

Would you like more details or have any questions about these representations?

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Here are 5 related questions that could deepen your understanding:

1. How would the interval notation change if \(x\) was allowed to be exactly between \(-2\) and \(0\)?
2. How can we represent a closed interval such as \([-2, 0]\) on the number line?
3. What is the interval notation for \(x\) being strictly less than \(-2\) or strictly greater than \(0\)?
4. How do you represent an intersection of two intervals using interval notation?
5. What is the difference between open and closed intervals, and how are they represented on the number line?

**Tip:** Use brackets \([]\) in interval notation to include endpoints and parentheses \(()\) to exclude endpoints.

Learn how to express the interval x ≤ -2 or x ≥ 0 using both interval notation and number line representations. Understand the concepts of inequalities, interval notation, and graphical representation on a number line. Suitable for students in Grades 9-12.

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