Problem 12 - College Algebra

Question

Express interval in set-builder notation and graph the interval on a number line. \((-\infty, 2)\)

Step-by-Step Solution

Answer

The set-builder notation of \((-\infty, 2)\) is \(x \mid x < 2\), and it is represented on a number line by an open circle at the number 2 with a line extending to the left towards negative infinity.

Step 1: Set-Builder Notation

To write the interval \((-\infty, 2)\) in set-builder notation, we denote it as: \(x \mid x < 2\). This means the set of all x such that x is less than 2.

Step 2: Graphical Representation

To graph \((-\infty, 2)\) on a number line, draw a line representing the real number line. Then, draw an open circle at the number 2 (because 2 is not included in the interval) and a line extending to the left from the circle indicating that the interval extends to negative infinity.

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers along a number line. It describes the continuous range between two endpoints.
In this exercise, the interval \((-fty, 2)\) means that the set includes all numbers less than 2, but not 2 itself.
Here’s how interval notation works:

• Parentheses **()** indicate that an endpoint is not included in the interval (open interval).
• Brackets **[]** show that an endpoint is included (closed interval).

For instance, \([a, b]\) includes all numbers from \(a\) to \(b\), including \(a\) and \(b\) themselves. Alternatively, \((a, b)\) includes numbers between \(a\) and \(b\), but not \(a\) or \(b\).
Understanding how to use these symbols is key for representing intervals accurately in mathematics.

Number Line Graph

A number line graph is a visual representation of numbers along a straight line. This helps you understand the range of values included in an interval.
To graph the interval \((-fty, 2)\) on a number line:

• Draw a straight line and mark several important numbers on it, like 0, 1, 2, 3, etc.
• Place an open circle at 2. This indicates that 2 is not included in the set.
• Draw a line extending to the left towards negative infinity. This shows all numbers less than 2 are part of the set.

Open and closed circles are crucial in these graphs. An open circle means the number is not part of the interval, while a closed circle would mean the number is included.

Inequalities

Inequalities are mathematical expressions that compare two values or expressions. They show the relative size of two values or how one value relates to another.
Here are some basic inequality symbols:

• \(<\) means "less than".
• \(>\) means "greater than".
• \(\leq\) means "less than or equal to".
• \(\geq\) means "greater than or equal to".

In the context of our exercise, the interval \((-fty, 2)\) is expressed in set-builder notation as \(x \mid x < 2\). This inequality "\(x < 2\)" means we're considering all numbers less than 2.
Understanding how to translate between interval notation and inequalities is essential for solving mathematical problems involving ranges of numbers.