Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ [4, \infty) $$

Step-by-Step Solution

Answer

The interval \([4, \infty)\) as an inequality is \(x \geq 4\).

Step 1: Identify the Interval Type

The interval given is \([4, \infty)\). This means it includes the number 4 and extends to positive infinity. It is a half-open interval including 4 and not including infinity.

Step 2: Convert to Inequality

For the interval \([4, \infty)\), we express this as an inequality by stating that the variable \(x\) is greater than or equal to 4. This is because the interval starts at 4 and goes towards infinity, meaning \(x\) takes all values from 4 onwards.

Step 3: Write the Final Inequality

Write the inequality for the expression: \(x \geq 4\). This captures all numbers from 4 to infinity, where \(x\) can be equal to or greater than 4.

Key Concepts

Inequalities

Inequalities are mathematical expressions used to compare two values. Instead of stating an exact equals relationship, inequalities show if one value is less than or greater than another. Frequently used symbols in inequalities include:

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

These symbols help define ranges of numbers for a given variable. In the context of intervals, inequalities are useful for expressing the set of numbers that lie within the boundaries of the interval. For example, when we have an interval like \([4, \infty)\), the inequality \(x \geq 4\) is used to represent all numbers starting from 4 and continuing towards infinity. This inequality tells us that \(x\) can take any value that is 4 or larger, emphasizing the concept of a boundless range starting from a specific point.

Half-Open Intervals

In mathematics, intervals define a specific range of numbers. A half-open interval is one where one endpoint is included in the interval, while the other is not. They are denoted using a combination of brackets and parentheses, where:

• A square bracket \([\) or \(]\) means the number at its end is included. It is also known as closed at that endpoint.
• A parenthesis \((\) or \()\) means the number at its end is not included, also known as open at that endpoint.

For example, the interval \([4, \infty)\) is half-open. Here:- The square bracket \([4\) indicates that 4 is included in the interval. This means any value of \(x\) must be equal to or greater than 4.- The parenthesis \(\infty)\) indicates that infinity is not a specific number we can include or touch, but represents an unbounded extent in that direction.Half-open intervals are crucial when expressing conditions where the variable can reach up to, but never actually reach, an open (unbounded) endpoint, like infinity.

Infinity in Mathematics

Infinity in mathematics represents a concept of something that is limitless or unbounded. Unlike ordinary numbers, infinity is not a quantity that can be measured or assigned a value. It is a notion used to describe quantities that are infinitely large or sequences that do not terminate.

• In the context of intervals, seeing infinity in \([4, \infty)\) represents an unending continuation from the number 4.
• Infinity is useful in mathematics to illustrate ideas in calculus, set theory, and beyond where values are not finite.
• Although labeled with the symbol \(\infty\), infinity cannot be included in a set or an interval like a typical number. This is why infinity's symbol is paired with a parenthesis \(()\) to express its unbounded nature.

Understanding that infinity represents an idea rather than a number is important in grasping how intervals function, especially when dealing with inequalities and half-open intervals.