Problem 50 - Algebra and Trigonometry

Question

Graph the set. $$ [-4,6) \cup[0,8) $$

Step-by-Step Solution

Answer

The graph is the interval \([-4,8)\).

Step 1: Understand the Interval Notation

The interval \([-4, 6)\) represents all numbers from \(-4\) to \(6\), inclusive of \(-4\) but not \(6\). The interval \([0, 8)\) represents all numbers from \(0\) to \(8\), inclusive of \(0\) but not \(8\). The union symbol \(\cup\) indicates that we take all points that belong to either interval.

Step 2: Determine the Combined Interval

To find the union of \([-4,6)\) and \([0,8)\), we need to combine the intervals in a way that includes all values covered by either interval. Starting from \(-4\), the intervals overlap and reach up to \(6\) from the first interval and continue to \(8\) because of the second interval, resulting in \([-4,8)\).

Step 3: Graph the Interval

On a number line, plot a closed dot at \(-4\) to indicate that \(-4\) is included. Draw a line extending from \(-4\) to \(8\) and place an open dot at \(8\) indicating it is not included.

Key Concepts

Interval Notation

Interval notation is a concise way of writing the set of all numbers between two endpoints. When using interval notation:

• Brackets "[ ]" are used to include an endpoint, meaning the endpoint is part of the interval.
• Parentheses "( )" are used to exclude an endpoint, meaning the endpoint is not part of the interval.

In the union problem, we work with the intervals \([-4, 6)\) which includes \(-4\) and excludes \(6\), and \([0, 8)\) which includes \(0\) and excludes \(8\).
The process of uniting these intervals involves combining all numbers that fall within either interval. The union \([-4, 6) \cup [0, 8)\) results in \([-4, 8)\).
This means any number greater than or equal to \(-4\) and less than \(8\) is included in the union.

Graphing on a Number Line

Graphing intervals on a number line provides a visual representation of which numbers are part of a set. Here's how to graph the union of \([-4, 6) \cup [0, 8)\):

• First, identify the numerical range for the intervals, which is \([-4, 8)\).
• Place a filled or closed dot at \(-4\) to show that it is part of the interval.
• Draw a continuous line extending from \(-4\) to just before \(8\), representing all the numbers included in the interval.
• Place an open dot at \(8\) to show this point is not included.

This graphical method allows you to easily see that every number from \(-4\) to before \(8\) is part of the set.

Set Theory

Set theory is a fundamental aspect of mathematics used to study collections of objects, called sets. The concept of a union is vital in set theory. Here, the union symbol \(\cup\) is used to denote the combination of two sets into one. When dealing with intervals in set theory:

• The union \(A \cup B\) consists of all elements that are in set \(A\), in set \(B\), or in both.
• In our problem, the union \([-4, 6) \cup [0, 8)\) results in a larger set that covers all numbers from \(-4\) through \(8\), excluding \(8\) itself.
• Set theory's notation and operations, like unions, are tools that help describe and analyze collections of objects or numbers succinctly.

Thus, understanding unions through set theory aids in comprehending how two intervals combine to form a single, inclusive interval.