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Interval notation is a way of representing a range of values for a variable. This can be particularly helpful for inequalities. Let's break down the two parts of the compound inequality given: \(x < -1 \) and \( x > 4 \).
\( x < -1 \) means that x includes all values less than -1, but not -1 itself. The notation for this is \((-\infty, -1)\).

\( x > 4 \) means x includes all values greater than 4, excluding 4 itself. The interval notation is \((4, \infty)\).

Since we are dealing with an 'or' condition here, we combine these two intervals with 'or': \((-\infty, -1) \text{ or } (4, \infty)\).

• Reminder: Use open parentheses \( ( ... ) \) when the endpoints are not included.
• Use brackets \( [ ... ] \) when the endpoints are included.

Understanding interval notation is key to interpreting and solving inequalities accurately.

Graphing inequalities is a method to visually represent the solutions of an inequality on a number line. For the compound inequality \( x < -1 \text{ or } x > 4 \), let's graph each part: \( x < -1 \) and \( x > 4 \).

• First, mark the points -1 and 4 on the number line.
• For \( x < -1 \), draw a leftward arrow starting at -1 and use an open circle to indicate that the point -1 is not included.
• For \( x > 4 \), draw a rightward arrow starting at 4 and use an open circle to show that 4 is excluded.

Combining these graphs on the same number line, we see two separate parts. This represents that x can be any value less than -1 or any value greater than 4, but not between -1 and 4.

A number line is a visual tool that helps us understand relationships between numbers and visualize solutions to inequalities. Here's how to use it for compound inequalities:

• Draw a horizontal line and evenly space out numbers along it.
• Find specific points, in this case, -1 and 4.
• For \( x < -1 \), place an open circle at -1 and shade or draw an arrow to the left.
• For \( x > 4 \), place an open circle at 4 and shade or draw an arrow to the right.

The number line now clearly shows the parts of the line where the inequality is true. Remember, open circles are used when the value is not included in the solution set.