Problem 24 - Elementary and Intermediate Algebra

Question

Graph each rational number on the number line. $$ -\frac{17}{5} $$

Step-by-Step Solution

Answer

Plot a point at \(-3.4\).

Step 1: Understand the Rational Number

A rational number is any number that can be expressed as the quotient or fraction of two integers. In this exercise, the given rational number is \(-\frac{17}{5}\).

Step 2: Convert the Fraction to a Mixed Number

Convert \(-\frac{17}{5}\) to a mixed number by dividing the numerator by the denominator. \(17 \div 5 = 3.4\), which means \(-\frac{17}{5} = -3\frac{2}{5}\).

Step 3: Identify the Points on the Number Line

Locate the integers on the number line around the mixed number. For \(-3\frac{2}{5}\), this will be between \(-3\) and \(-4\).

Step 4: Divide the Interval Between Integers

Divide the interval between \(-3\) and \(-4\) into 5 equal parts because the denominator is 5. Each part represents \(-0.2\).

Step 5: Plot the Rational Number

Count 2 parts to the left of \(-3\), because \(-3\frac{2}{5}\) means 2 parts of \(-0.2\). Therefore, plot a point at \(-3.4\) on the number line.

Key Concepts

number line

A number line is a visual representation of numbers placed in order on a straight horizontal line. Each point on the line corresponds to a number, and these numbers can be positive, negative, or zero. The numbers to the right of zero are positive, and the numbers to the left of zero are negative. The number line helps us to easily see the relative sizes of numbers and understand their order.

When working with rational numbers, visualizing them on a number line can help us to better understand their placement and relationship to other numbers. By dividing intervals into equal parts based on the fraction's denominator, we can accurately plot these numbers.

mixed numbers

Mixed numbers include both a whole number and a fraction. For example, \(-3\frac{2}{5}\) is a mixed number made of the whole number \(-3\) and the fraction \(-\frac{2}{5}\). Converting fractions to mixed numbers makes it easier to locate their position on the number line.

To convert a fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder over the original denominator becomes the fractional part. In our example, \(-\frac{17}{5}\) becomes \(-3\frac{2}{5}\). By understanding mixed numbers, we can segment and visualize them more accurately on the number line.

rational numbers

Rational numbers are numbers that can be written as the quotient of two integers. In other words, any number that can be expressed in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is not zero, is a rational number. Examples of rational numbers include: \(-\frac{17}{5}\), \(\frac{2}{3}\), and \(-1\).

Rational numbers can also be either positive or negative. Negative rational numbers, like \(-\frac{17}{5}\), imply that either the numerator or the denominator is negative, but not both. Understanding rational numbers is essential for proper graphing and for recognizing their positions relative to other numbers on a number line.

fractions

A fraction represents a part of a whole and is written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. The denominator indicates into how many equal parts the whole is divided, while the numerator shows how many of those parts we have. For instance, in \(-\frac{17}{5}\), 5 is the denominator suggesting the whole is split into 5 parts, and 17 is the numerator indicating we have 17 of those parts.

Fractions can be greater than, less than, or equal to one, and they can also be translated into decimals. By understanding how fractions work, we can convert them to mixed numbers and easily plot them on the number line. Dividing the number line into parts based on the denominator gives us precise points for plotting the fractions.