Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Give a verbal description of the subset of real numbers that is represented by the inequality, and sketch the subset on the real number line.\(x \geq-2\)

Step-by-Step Solution

Answer

The subset of real numbers represented by the inequality \(x \geq -2\) consists of -2 and all real numbers greater than -2.

Step 1: Interpret the Inequality

Here, the inequality \(x \geq -2\) represents all real numbers x that are greater than or equal to -2. That is, it includes -2 and all real numbers to the right of -2.

Step 2: Sketch the Subset on the Real Number Line

Draw a number line and mark -2 on it. A solid dot is placed on -2 to represent that -2 is included in the subset. From -2, draw an arrow pointing to the right to represent all the real numbers greater than -2.

Key Concepts

Number Line Sketch

A number line sketch is a powerful tool used to visualize relationships between numbers, especially when dealing with inequalities. It involves drawing a horizontal line which represents all possible values from negative to positive infinity. When sketching the subset of real numbers for an inequality such as \(x \geq -2\), you start by identifying the critical number—here, it's -2. On your number line, this number will be your reference point. You mark it, often with a labeled tick mark.

To illustrate the inequality \(x \geq -2\), which includes all numbers greater than or equal to -2, you draw an arrow extending from -2 to the right side of the number line. This arrow captures the idea that every number beyond -2 is a part of this subset, without any upper limit. By sketching on a number line, you can quickly see the full range of numbers that satisfy the inequality.

Inequality Interpretation

The interpretation of an inequality is fundamental for understanding the relationships it represents among real numbers. When we are given an inequality like \(x \geq -2\), it explains how two expressions are related. In this case, the inequality is saying that the variable \(x\) is either exactly -2 or any number larger than that.

In plain language, this means we're looking at a range that starts at -2 and stretches to infinity. Every real number within this range, no matter how large, is a solution to the inequality. This interpretation allows us to understand the set of all possible solutions as a continuous set rather than just individual numbers.

Real Numbers Subset

The subset of real numbers is a specific portion of the continuous range of all real numbers that satisfy a given condition. In our case, the condition is the inequality \(x \geq -2\). The subset that corresponds to this inequality is the set of all real numbers that are greater than or equal to -2.

Visualizing the Subset

One can think of this subset as a segment of the entire number line. It has a starting point, which is the number -2, but it has no endpoint because it stretches infinitely to the right. When communicated in set notation, it is written as \([-2, \infty)\), indicating a closed interval at -2 and an open interval at infinity, since infinity is not a number that can be reached or included.

Solid Dot Notation

Solid dot notation is a specific way of marking a number line to show that a particular value is included in the set of solutions to an inequality. When we have an inequality like \(x \geq -2\), the solid dot is placed exactly at -2 on the number line. This is crucial because the solid dot communicates inclusion; it signals that -2 itself is part of the solution set.

Why the Solid Dot Matters

Had the inequality been \(x > -2\) instead, an open dot would be used at -2, showing that -2 is not included in the solution set. It's essential to differentiate between these notations because they represent distinctly different scenarios in terms of which numbers belong to the subset defined by the inequality.