Problem 102 - Prealgebra and Introductory Algebra

Question

Place the correct symbol, \(<,=,\) or \(>,\) between the two numbers. $$|7| \quad|-9|$$

Step-by-Step Solution

Answer

\(|7|<|-9|\)

Step 1: Compute the absolute value of 7

The absolute value of a number is its distance from 0 on the number line. Since 7 is 7 units away from 0, its absolute value is 7. So, \(|7|=7\)

Step 2: Compute the absolute value of -9

Using the same logic, the absolute value of -9 is its distance from 0 on the number line. This is also 9 units away from 0, so the absolute value of -9 is 9. Therefore, \(|-9|=9\)

Step 3: Compare the two computed values

Now compare the absolute values. 7 is less than 9. So the correct symbol to put between these two numbers is '<'.

Key Concepts

Comparing Numbers

Understanding how to compare numbers is a fundamental mathematical skill. When we compare two numbers, we are determining whether one is greater than, less than, or equal to the other. This is typically done using the inequality symbols: `<` (less than), `=` (equals), and `>` (greater than).

For example, the statement \(7 < 9\) is read as '7 is less than 9'. It tells us that if we were to place the numbers on a scale or a number line, 7 would be to the left of 9, indicating it has a lower value. This concept becomes slightly more intricate when dealing with absolute values, as they require us to consider the distance of a number from zero rather than its position as a positive or negative number on the number line.

Number Line

The number line is a visual representation that helps us understand the position and magnitude of numbers. It's like a horizontal ruler with zero in the middle, positive numbers to the right, and negative numbers to the left. Each point on the line corresponds to a number, and the distance of the point from zero represents the absolute value of that number. It helps us easily compare numbers and comprehend operations like addition, subtraction, and identifying absolute values.

When visualizing the problem \( |7| \quad |-9| \) on a number line, we'd see two points: one at positive 7 and another at negative 9. Despite the negative sign, the absolute value dictates we only consider the distance from zero, so both points are equally distant but on opposite sides of zero.

Properties of Absolute Value

The absolute value of a number can be thought of as its distance from zero on a number line without regard for direction. This means that the absolute value is always non-negative. The notation for absolute value is two vertical lines surrounding a number, for example, \( |7| \).

Here are several key properties of absolute values:

• The absolute value of zero is zero (\( |0| = 0 \)).
• The absolute value of a positive or negative number is always positive (\( |-9| = 9 \), \( |7| = 7 \)).
• The absolute value of a product is the product of the absolute values (\( |ab| = |a||b| \)).
• The absolute value of a quotient is the quotient of the absolute values (\( \left|\frac{a}{b}\right| = \frac{|a|}{|b|} \), given \( b \eq 0 \)).

The properties greatly simplify the comparison and arithmetic of numbers by reducing the complexity involved when dealing with positive and negative values.