Problem 34 - 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.\(0 \leq x \leq 5\)

Step-by-Step Solution

Answer

The subset of real numbers represented by the inequality \(0 \leq x \leq 5\) includes all real numbers between 0 and 5, inclusive. These numbers are represented on the number line as a filled circle at 0 and at 5, and a line connecting these two points.

Step 1: Interpret the Inequality

The inequality \(0 \leq x \leq 5\) is describing a subset of the real numbers. The symbol \(\leq\) means 'less than or equal to'. Therefore, this inequality is saying that x is a real number that is greater than or equal to 0 and less than or equal to 5.

Step 2: Verbal Description

So the verbal description of the subset described by the inequality is: all the real numbers that are between 0 and 5, inclusive.

Step 3: Draw the Subset on the Number Line

Draw a number line. Mark a point for 0 and a point for 5. Because the inequality includes 0 and 5 (shown by the \(\leq\) symbol), draw a filled circle at these points. Draw a line between these two points to show that all numbers between 0 and 5 are included in the subset.

Key Concepts

Number Line

The number line is a visual representation used to understand the position of real numbers in a linear sequence. It is an infinite line with numbers placed at convenient intervals, typically marked by evenly spaced points.
In mathematics, it helps us easily interpret data, such as inequalities.In the given exercise, when you draw a number line for the inequality \(0 \leq x \leq 5\), you mark the number 0 and the number 5 on the line. You also note all numbers between these two points. This visual representation is crucial for identifying and understanding subsets of real numbers. By using filled circles at 0 and 5, we indicate that these values are included in the subset, as specified by the inequality.

Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes rational numbers, like fractions and whole numbers, as well as irrational numbers, like \(\pi\) and \(\sqrt{2}\). Real numbers make up the entirety of the numerical spectrum we often work with.In this exercise's context, when we talk about the subset of real numbers between 0 and 5, it means we consider any number that can be pinpointed on the number line within these bounds. The bounds are set by the inequality \(0 \leq x \leq 5\), which includes all real numbers starting from 0 and ending at 5. All these numbers, whether 2.5, 3, or even 4.999, fall within this category.

Graphical Representation

Graphical representation is a powerful way to understand data and mathematical concepts visually. It allows us to see and interpret the relationships and variations within data sets at a glance.In this exercise, by representing the inequality \(0 \leq x \leq 5\) on a number line, we achieve a graphical understanding of the subset of real numbers.
The filled circles at 0 and 5, coupled with the line connecting them, make it easy to see which numbers are included in the subset. It highlights how the inequality encompasses all numbers from 0 to 5, reinforcing the concept that within this range, every point on the line represents a real number. This simple line graph is an efficient way to interpret and convey findings in mathematics.