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).