Problem 69 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$\left|-\frac{9}{7}\right|$$

Step-by-Step Solution

Answer

\(\left| -\frac{9}{7} \right| = \frac{9}{7}\)

Step 1 - Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always positive or zero.

Step 2 - Identify the Number

Here, the number in question is \(-\frac{9}{7}\).

Step 3 - Remove the Negative Sign

Since absolute value disregards the sign, \(\left| -\frac{9}{7} \right| = \frac{9}{7}\).

Key Concepts

Number Line

A number line is a visual representation of numbers arranged in order. It helps you understand the position of both positive and negative numbers. Imagine drawing a straight line on paper:

• To the right of 0, you have positive numbers (1, 2, 3, etc.).
• To the left of 0, you have negative numbers (-1, -2, -3, etc.).
• Fractions and decimals also fit on the number line. For example, 0.5 is between 0 and 1, while -0.5 is between -1 and 0.

The absolute value of a number is its distance from 0, so it is always a positive number or zero. You can use the number line to visualize this. For instance, the absolute value of -3 is 3 because it is three steps away from 0. This concept is essential for understanding why \(\left| -\frac{9}{7}\right|\) equals \(\frac{9}{7}\).

Positive Numbers

Positive numbers are numbers greater than zero. They are written without any sign or with a plus sign (such as +5 or simply 5). Uses of positive numbers include counting objects, measuring distances, and financial gains.
In the context of absolute value, a positive number has the same absolute value as the number itself. This is why we say \(\left| 7 \right| = 7\). No matter the context, the concept of positivity helps in understanding that absolute values disregard the direction or sign of a number entirely.
For example:

• The absolute value of +2 is 2.
• The absolute value of \-\frac{9}{7}\ is \(\frac{9}{7}\).
Let's apply this to fractions to understand their positive value regardless of the initial sign.

Fractions

A fraction represents a part of a whole. It consists of a numerator, the top number, and a denominator, the bottom number. For example, in \(\frac{9}{7}\), 9 is the numerator, and 7 is the denominator.
Fractions can be positive or negative. The sign of a fraction depends on its components. If the numerator or denominator is negative, the fraction is negative, like \(\frac{-9}{7}\). If both are positive, the fraction is positive, like \(\frac{9}{7}\).
When taking the absolute value of a fraction, you find its positive equivalent. For instance, the absolute value of \(\frac{-9}{7}\) is \(\frac{9}{7}\).
This makes it easier to work with these numbers in various math problems. Remember:

• A positive fraction stays the same.
• A negative fraction loses its negative sign.