Problem 15 - College Algebra

Question

Express interval in set-builder notation and graph the interval on a number line. \((-3,0) \cap[-1,2]\)

Step-by-Step Solution

Answer

The set-builder notation for the given interval can be expressed as \(-1 \leq x < 0\). The graphical representation of the interval on the number line will have a filled circle at -1 and an open circle at 0.

Step 1: Understand the notation

It is important to understand that the parenthesis such as (a,b) denotes an open interval which includes all real numbers from a to b but does not include the numbers a and b themselves, whereas the brackets [a,b] denote a closed interval which includes all real numbers from a to b including the numbers a and b.

Step 2: Intersect the intervals

Since the task is to find the intersection (\(\cap\)) of the intervals (-3,0) and [-1,2], we must find the numbers that are common in both the intervals. In this case, the common numbers are all values from -1 to 0, but 0 is not included in the first interval (-3,0). So we have the intersection as [-1,0).

Step 3: Convert to set-builder notation

To convert this to the set-builder notation, we express it as the set of all real numbers x such that x is greater than or equal to -1 and less than 0. In set-builder notation: \(-1 \leq x < 0\).

Step 4: Draw on Number line

For the graphical representation, represent the interval [-1,0) on the number line. Start by marking -1 and 0 on the number line. Since -1 is included in the interval, represent it as filled circle (●) and as 0 is not included, represent it as open circle (○).

Key Concepts

Interval Notation

Interval notation is a shorthand used in mathematics to describe a set of numbers that fall within an interval. It uses parentheses and brackets to indicate whether endpoints are included or not.

• Parentheses "(a, b)" denote an open interval, meaning that the endpoints "a" and "b" are not included.
• Brackets "[a, b]" denote a closed interval, meaning that the endpoints "a" and "b" are included.

This notation simplifies the expression of complex sets and is widely used in calculus and algebra to convey information regarding solutions to inequalities and domains of functions.

Intersection of Intervals

The intersection of intervals involves finding the set of numbers that are common to two or more intervals. It is denoted by the symbol \(\cap\).

• To find the intersection, look at the overlapping part of the number lines represented by each interval.
• Only numbers that lie within both intervals are part of the intersection.

For example, the intersection of the intervals \((-3,0)\) and \([-1,2]\) is \([-1, 0)\), as this is where the two intervals overlap on the number line.

Number Line Representation

Number line representation is a visual method to depict intervals, making it easier to understand and interpret mathematical solutions.

• Start by marking key points on the number line, such as endpoints of intervals.
• Visual symbols such as open circles (○) and closed circles (●) are used to indicate whether endpoints are included or not.

For instance, the interval \([-1, 0)\) is illustrated on a number line with a filled circle at "-1" (included) and an open circle at "0" (not included). This approach simplifies the process of identifying overlaps and intersections between intervals.

Open and Closed Intervals

Open and closed intervals help us understand which portions of the interval are included in the set. This distinction is crucial in solutions involving inequalities.

• Open intervals, indicated by parentheses, do not include their endpoints. For example, \((a, b)\) means that neither "a" nor "b" are part of the interval.
• Closed intervals, indicated by brackets, include their endpoints. For example, \([a, b]\) means "a" and "b" are included in the interval.

This concept is especially important when solving inequalities, as it guides whether boundary values satisfy the conditions of the solution.