Problem 13 - Algebra and Trigonometry

Question

In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. $$ (-\infty, 5.5) $$

Step-by-Step Solution

Answer

The set-builder notation for the interval \(-\infty, 5.5)\) is \{x | x \in R, x < 5.5\}. On a number line, the interval extends from 5.5 to the left with an open circle at 5.5.

Step 1: Convert to Set-Builder Notation

Set-builder notation describes a set using a property that the elements of the set have in common. In this case, the common property is that the numbers are less than 5.5 but do not include 5.5. The set-builder notation is therefore: \{x | x \in R, x < 5.5\}

Step 2: Graph the Interval on a Number Line

For graphing, draw a number line, mark a point for the number 5.5, and draw a line from 5.5 to the left (since the numbers are less than 5.5). To indicate that 5.5 is not included in the interval, an open circle should be used at 5.5.

Key Concepts

Interval Notation

Interval notation is a way of describing a set of numbers by specifying the smallest and largest numbers in the interval. It's a common language in mathematics used to express ranges of numbers concisely. For example,

• \((-\infty, 5.5)\) describes all numbers less than 5.5. The parenthesis means 5.5 is not included in the set.
• If both ends of an interval are included, brackets [ ] are used.

Intervals with positive or negative infinities always use parentheses because infinity is a conceptual idea, not a number.
Using interval notation helps you see at a glance where numbers begin and end in a set. It's a handy tool for conveying complex number ranges simply.

Number Line Graphing

Graphing on a number line is a visual way to represent numbers and intervals. It helps to see what set of numbers we are considering. For instance:

• Draw a straightforward line called a number line.
• Mark the specific number you are interested in, such as 5.5 in this case.
• To represent numbers less than 5.5, draw an arrow extending to the left from 5.5.
• An open circle is used at 5.5 to show that it is not included in the set.

This method allows you to quickly understand which numbers are part of the set. It's excellent for visual learners who benefit from seeing things drawn out.

Real Numbers

Real numbers include all the numbers you know on the number line: whole numbers, fractions, decimals, and irrational numbers. They form the backbone of most number systems used in mathematics.

• Whole numbers like 1, 2, 3.
• Fractions such as 1/2 or 3/4.
• Decimals like 2.5 or -0.75.
• Irrational numbers such as \(\sqrt{2}\) or \(\pi\).

Real numbers are everywhere in math and science because they help in measuring and understanding the world. They fill up the entire number line without gaps, which is why they are so essential in everyday calculations.

Open Intervals

An open interval indicates a range of numbers that does not include its endpoints. Consider the interval \((-\infty, 5.5)\):

• The parentheses around 5.5 mean that 5.5 is not part of the set.
• This type of interval is called "open" because both ends (or one end) are excluded.

Open intervals are used when you're interested in numbers within a boundary set, but not necessarily at the boundary itself. They can be contrasted with closed intervals, which include the boundaries denoted by square brackets \([ ]\). Understanding these differences is crucial when working with inequalities or ranges in various math problems.