Problem 59 - College Algebra

Question

Rewrite expression without absolute value bars. \(|-3|-|-7|\)

Step-by-Step Solution

Answer

The expression \( |-3|-|-7| \) without absolute value bars is -4.

Step 1: Solve Absolute Value of -3

For the first part \( |-3| \), the absolute value of -3 simply refers to its distance from zero on a number line. Since the distance is only recorded in positive terms, \( |-3| \) equals 3.

Step 2: Solve Absolute Value of -7

Next, for the part \( |-7| \), the distance of -7 from zero on a number line is 7 when counted in positive terms, so \( |-7| \) equals to 7.

Step 3: Subtract Step 2 from Step 1

Finally, subtract the absolute value of -7 from the absolute value of -3. So, 3 - 7 equals -4.

Key Concepts

Understanding Absolute Value

The concept of absolute value is a fundamental part of mathematics, particularly when dealing with distances and the idea of magnitude regardless of direction. In simple terms, the absolute value of a number signifies its distance from zero on a number line. This distance is always expressed as a positive number or zero, as it doesn't have a direction.

For instance, consider the absolute value of -3, represented as \(|-3|\). On a number line, whether you move 3 units to the left or to the right from zero, the distance you cover is the same: 3 units. Similarly, the expression \(|-7|\) indicates the distance of -7 from zero, which is also a positive 7, despite -7 being located in the opposite direction from 3.

The Role of the Number Line

A number line is a visual representation that helps us grasp the concept of numbers and their magnitudes. It's a straight line with numbers placed at equal intervals along its length. Positive numbers are situated to the right of zero, while negative numbers are to the left. When it comes to absolute value, the number line is particularly helpful because it allows us to see that the absolute value of any number is its distance from zero on this line.

Using the earlier example of \(|-3|\) and \(|-7|\), mapping these numbers on a number line clearly shows that -3 is three spaces to the left of zero, while -7 is seven spaces to the left. Counting these spaces will always give you a positive number; hence, \(|-3| = 3\) and \(|-7| = 7\). This also helps to understand subtraction and addition of absolute values in terms of distances and directions on the number line.

Algebraic Expressions Without Absolute Value

Algebraic expressions can sometimes include absolute values, which can complicate the simplification process. To simplify such expressions, you must first understand the absolute value before applying arithmetic operations. In the given exercise, the expression \(|-3|-|-7|\) first requires us to evaluate the absolute values independently, then perform the subtraction.

As established, the absolute value of -3 is 3, and the absolute value of -7 is 7. Once these are determined, the expression boils down to a basic algebraic operation of subtraction: \(3 - 7\), yielding the final simplified result of -4. This result is free of absolute value bars and reflects the net distance between the points represented by \(|-3|\) and \(|-7|\) on the number line.