Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-20: Express the interval as an inequality in the variable \(x\). $$ (-\infty,-5) $$

Step-by-Step Solution

Answer

The inequality is \(x < -5\).

Step 1: Identify the Interval Type

The given interval is \((-\infty, -5)\), which means it includes all real numbers less than \(-5\). The round bracket indicates that \(-5\) is not included in the interval.

Step 2: Interpret the Interval as an Inequality

Since the interval represents numbers less than \(-5\), we can express this as an inequality: \(x < -5\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It succinctly expresses the set of all numbers lying within a specific range. This is particularly useful to handle inequalities efficiently. For instance, consider the interval \((-\infty, -5)\). Here, the parentheses \(()\) mean that the endpoint \(-5\) is not included in the interval, reflecting an 'open' interval.
This notation can be broken down into different components:

• \( -\infty \) signifies that the interval starts from negative infinity, indicating no lower bound exists.
• The symbol \(-5\) is the upper boundary of the interval.

Open intervals such as these play an essential role in many areas of mathematics and are an efficient way to represent solutions for inequalities.

Real Numbers

Real numbers encompass a complete and continuous set of numbers that include both rational and irrational numbers. They form the backbone of many mathematical operations and concepts. In the context of the given interval, all values included are part of this set of real numbers.

• **Rational numbers**: These are numbers that can be expressed as fractions, such as \(3/2\) or \(-1\).
• **Irrational numbers**: Numbers that cannot be represented as simple fractions, like \(\sqrt{2}\) or \(\pi\).

When dealing with an interval such as \((-\infty, -5)\), it includes every real number that is less than \(-5\). This selection excludes imaginary numbers, focusing purely on the real number subset.

Algebra

Algebra offers a systematic way of solving problems, including those involving inequalities. The original exercise asks us to express an interval in the form of an inequality. To achieve this, algebra comes into play as it provides the tools and rules required to transform expressions.
The give interval \((-\infty, -5)\) can be translated into an inequality using simple algebraic rules:

• Recognize that \(-5\) is the boundary and is not included, likely due to the parenthesis \())\).
• Conclude that every number less than \(-5\) must satisfy this inequality, hence we write \(x < -5\).

This algebraic manipulation allows a more intuitive understanding of such mathematical expressions and is fundamental to correctly translate between interval notation and inequalities.