Express the interval using two different representations.

x, is less than, minus, 6, or , x, is greater than, minus, 1

*x*<−6 or *x*>−1

Answer

Attempt 1 out of 2

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Interval Notation:

Number Line:

To express the interval \( x < -6 \) or \( x > -1 \) using two different representations, we will use **interval notation** and a **number line**.

### 1. Interval Notation

The given conditions are:
1. \( x < -6 \)
2. \( x > -1 \)

In interval notation:

- \( x < -6 \) is written as \( (-\infty, -6) \).
- \( x > -1 \) is written as \( (-1, \infty) \).

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:

\[
(-\infty, -6) \cup (-1, \infty)
\]

### 2. Number Line Representation

On a number line:

- We have an open circle at \( x = -6 \) and shade all values to the left (towards negative infinity) to represent \( x < -6 \).
- Similarly, we have an open circle at \( x = -1 \) and shade all values to the right (towards positive infinity) to represent \( x > -1 \).

The number line would look like this:

```
<-----o-----------------o----->
    -6                 -1
```

Where:
- The left part from \( -6 \) extending to \( -\infty \) is shaded.
- The right part from \( -1 \) extending to \( \infty \) is also shaded.

### Summary
- **Interval Notation:** \((- \infty, -6) \cup (-1, \infty)\)
- **Number Line:** Shaded regions to the left of \(-6\) and to the right of \(-1\) with open circles at \(-6\) and \(-1\).

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 also allowed to be exactly \(-6\) or \(-1\)?
2. How can we represent a closed interval on the number line?
3. What is the interval notation for \( x \) being between \(-6\) and \(-1\), inclusive?
4. How do you represent an intersection of two intervals using interval notation?
5. How would you express \( x \) being between two values, such as \( -6 < x < -1 \)?

**Tip:** Always use parentheses \(()\) for intervals that do not include endpoints (open intervals) and brackets \([]\) for intervals that include endpoints (closed intervals).

Learn how to express interval inequalities such as x < -6 or x > -1 using interval notation and number line representation. This problem covers key concepts in inequalities and interval notation, suitable for students in Grades 9-12.

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