Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ (-3, \infty) $$

Step-by-Step Solution

Answer

The inequality is \(x > -3\).

Step 1: Understand the Interval Notation

The interval \(-3, \infty)\) represents all numbers starting from just above \(-3\) and extending to infinity. It does not include \(-3\) itself.

Step 2: Convert to Inequality

In inequality notation, an open interval from \(-3\) to infinity means that the numbers are greater than \(-3\). This inequality is written as \(x > -3\).

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers that fall between a specified range. It often involves using parentheses or brackets to indicate whether the endpoints of the interval are included or not. For instance, the interval

• (a, b) uses parentheses to show an open interval, meaning that the endpoints a and b are not included.
• [a, b] uses brackets to show a closed interval, where both endpoints are included.
• [a, b) or (a, b] indicate a half-open or half-closed interval, meaning one endpoint is included while the other is not.

Interval notation is a concise way to convey this information without using inequality symbols. For example,

• (-3,
  ∞) represents all numbers greater than -3, going up infinitely, without including -3.

Open Interval

An open interval is one where neither of the endpoints are included in the set. This is represented in interval notation with parentheses. For example, the open interval (a, b) includes all numbers between a and b but does not include a and b themselves. Other examples include:

• The interval
  (-3,
  5) which includes all the numbers between -3 and 5 but not those exact numbers.
• The interval (0, ∞), indicating all positive numbers greater than 0.

When you encounter an open interval, it is important to remember that it consists of numbers strictly between the given endpoints. This concept helps set boundaries without allowing the endpoints to be part of the set.

Inequality Notation

Inequality notation uses symbols to express the range of values that a variable can take for a particular solution. Common inequality symbols include:

• "<" meaning less than
• ">" meaning greater than
• "≤" meaning less than or equal to
• "≥" meaning greater than or equal to

To convert an interval notation into inequality notation, make sure to pay attention to whether the interval is open or closed. For instance, the interval (-3, ∞) can be expressed in inequality notation as

• 
  x > -3, meaning x can be any number greater than -3.

By performing this conversion, you can more easily integrate solutions into equations and inequalities, which are frequently used in math problems.