Problem 2 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ x<100 $$

Step-by-Step Solution

Answer

(-\infty, 100)

Step 1: Understand Inequalities

The inequality \( x < 100 \) includes all real numbers that are less than 100. It is important to remember that this does not include 100 itself, as indicated by the "less than" sign without an "equal to" part.

Step 2: Identify the Interval

Since the inequality \( x < 100 \) represents all numbers less than 100, it extends from negative infinity up to but not including 100. The endpoint at 100 is open, meaning it is not included in the interval.

Step 3: Write the Interval Notation

In interval notation, we express the solution set for \( x < 100 \) as \( (-\infty, 100) \), using a parenthesis to show that 100 is not part of the interval and using \(-\infty\) because the numbers extend indefinitely in the negative direction.

Key Concepts

Inequalities

Inequalities are expressions or equations that compare two values, indicating one is either less than, greater than, or possibly equal to another. In our example, we have the inequality \( x < 100 \). This means that \( x \), our variable, can be any number that is less than 100. Since there's no "equal to" part, 100 itself is not included. These types of expressions are common in mathematics and help us define ranges or sets of numbers that satisfy the condition given by the inequality sign.
To solve inequalities:

• Identify the relationship between the variable and the number in question.
• Determine if the inequality includes the number itself ("\(\leq\)" or "\(\geq\)") or excludes it ("\( < \)" or "\( > \)").

In our case, "\( < \)" tells us 100 isn't part of the solution set.

Real Numbers

Real numbers encompass nearly all possible numbers you can think of on the number line. They include rational numbers like fractions and integers, as well as irrational numbers like \( \pi \) and the square root of 2. When dealing with inequalities such as \( x < 100 \), we're talking about real numbers as our possible solutions. This is because real numbers are infinitely dense; between any two numbers, there's always another real number.
Understanding real numbers is key:

• Rational numbers: numbers that can be expressed as the quotient of two integers, such as 0.5 or -2.
• Irrational numbers: numbers that cannot be written as a simple fraction, such as \( \sqrt{3} \) or \( e \).

Real numbers give us a vast universe of values \( x \) can assume, stretching from infinitely negative to any number just shy of 100.

Solution Set

The solution set of an inequality or equation is the collection of all possible values that satisfy the expression. For the inequality \( x < 100 \), this set consists of every real number less than 100. To express the solution set clearly and concisely, we use interval notation.
Interval Notation is:

• A mathematical shorthand used to represent all numbers between a given range.
• It's written with parentheses or brackets, indicating an open (not included) or closed (included) endpoint.

The interval for \( x < 100 \) is written as \((-\infty, 100)\). The use of parenthesis around 100 signifies that 100 isn't part of the solution set, while "\(-\infty\)" shows that the numbers don't stop, continuing indefinitely in the negative direction.