91Ó°ÊÓ

When working with intervals in mathematics, we use interval notation to describe a range of values. It’s a shorthand to represent all numbers between two endpoints.
For instance, the interval \((-\infty, 5)\) includes all real numbers less than 5, but not 5 itself. We use a parenthesis at 5 to indicate that 5 is not part of the interval.
In contrast, the interval \([-3, \infty)\) includes all real numbers starting from -3 onwards, including -3 itself.
We use a square bracket at -3 to show that -3 is included.
Understanding interval notation is key to grasping further concepts like the union and intersection of intervals.

The union of intervals is a way to combine two or more intervals. The union includes all the values that are in either of the intervals.
Let's look at the union of \((-\infty, 5)\) and \([-3, \infty)\):
The first interval spans from negative infinity up to, but not including, 5.
The second one starts from -3 and stretches to positive infinity, including -3. When we take the union, we include all the values from both intervals.

• This means all numbers less than 5 (from the first interval).
• It also means all numbers starting from -3 to positive infinity (from the second interval).

Since the intervals overlap, the union covers every number from negative infinity to positive infinity.

Overlapping intervals occur when two or more intervals share some common values. In our example, \((-\infty, 5)\) and \([-3, \infty)\) have overlapping sections.
The first interval includes numbers up to, but not including, 5.
The second interval starts from -3 and goes into positive infinity.
The overlapping part is from -3 to 5. However, considering that \([-3, \infty)\) goes beyond 5 to infinity, we must combine all values.
Since the first interval also covers values left of -3 to negative infinity, combining both gives us \((-\infty, \infty)\).
In general, understanding overlapping intervals helps in correctly identifying regions of combination and solving union or intersection problems efficiently.