Problem 16 - College Algebra

Question

Express interval in set-builder notation and graph the interval on a number line. \((-4,0) \cap[-2,1]\)

Step-by-Step Solution

Answer

The intersection of intervals \((-4,0)\) and \([-2,1]\) is \([-2,0)\), which in set-builder notation is \(\{x|-2 \leq x < 0\}\).

Step 1: Analyze the intervals

Firstly, let's observe the given intervals. The first interval is \((-4,0)\), which represents all numbers between -4 and 0, not including -4 and 0. The second interval is \([-2,1]\), which represents all numbers between -2 and 1, including -2 and 1.

Step 2: Find the intersection

Next, let's find the intersection of these two intervals, which is the common part of these two sets of numbers. The interval that falls within both \((-4,0)\) and \([-2,1]\) is \([-2,0)\), which represents all numbers between -2 and 0, including -2 and not including 0.

Step 3: Convert to set-builder notation

Now, let's convert the interval into the set-builder notation. The interval \([-2,0)\) can be written as \(\{x|-2 \leq x < 0 \)\}, which means the set of all x such that x is greater than or equal to -2 and less than 0.

Step 4: Graph the interval on a number line

Lastly, we'll graph the interval on a number line. Draw a number line starting from -4 and ending at 1. A closed dot at -2 is drawn to indicate that -2 is included in the interval and an open dot at 0, to indicate that 0 is not included.

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way of describing a set by stating the properties its elements must satisfy. It is particularly useful in mathematics for defining intervals and more complex sets. The general form is \( \{ x \mid \text{condition} \} \), where the condition describes a range or specific properties of \( x \). In our example, the interval \([-2, 0)\) is expressed in set-builder notation as \( \{ x \mid -2 \leq x < 0 \} \). This means we include every number \( x \) where \( x \) is greater than or equal to \(-2\) and less than \(0\).

Key features of set-builder notation:

• Braces \( \{ \} \): Indicate that we are defining a set.
• The bar \( \mid \): Means "such that" and introduces the condition.
• Condition: Specifies the rule that elements of the set need to satisfy.

This notation simplifies how we present solutions in mathematics, especially for intervals and inequalities.

Number Line Graphing

Number line graphing is a visual way to represent intervals or numbers. It makes it easier to understand the relationship between different numbers and intervals. When graphing intervals, we use dots and lines to show which numbers belong to the interval.

For example, graphing the interval \([-2, 0)\) on a number line, we do the following:

• Closed dot at -2: Indicates that \(-2\) is included in the interval.
• Open dot at 0: Shows that \(0\) is not part of the interval.
• Line segment: Connects these indicators, covering all numbers between and including \(-2\) and less than \(0\).

Number line graphs are excellent for visualizing how intervals overlap or otherwise relate to each other, serving as a useful tool for finding intersections or unions of sets.

Intersection of Intervals

The intersection of intervals involves finding the common elements that lie in all given sets or intervals. It is similar to finding the overlap or common area shared by different intervals. The intersection tells us which values satisfy the conditions of all intervals involved.

In the exercise, we have the intervals \((-4, 0)\) and \([-2, 1]\). To find their intersection:

• Start point: Take the larger of the two interval start points, which is \(-2\).
• End point: Take the smaller of the two interval end points, which is \(0\), but not including \(0\) since \(0\) is an open end in \((-4, 0)\).
• Result: The intersection is \([-2, 0)\).

Thus, the intersection \([-2, 0)\) includes all numbers that both original intervals share, clearly displayed through number line graphing.