Problem 51 - Fundamentals of Mathematics

Question

Determine each value. $$ |-5|+|-2| $$

Step-by-Step Solution

Answer

The value is 7.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Essentially, it makes any number positive. Therefore, for a number \( x \), the absolute value \( |x| \) is \( x \) if \( x \geq 0 \) and \(-x\) if \( x < 0 \).

Step 2: Apply Absolute Value to -5

Since \(-5\) is less than zero, we take its opposite to find the absolute value. Thus, \( |-5| = 5 \).

Step 3: Apply Absolute Value to -2

Similarly, since \(-2\) is less than zero, we take its opposite. Therefore, \( |-2| = 2 \).

Step 4: Add the Absolute Values

Now, add the absolute values obtained: \( 5 + 2 = 7 \).

Key Concepts

Number Line

A number line is a straight, horizontal line used to visually represent numbers. Each point on this line corresponds to a number, with zero typically positioned at the center.

- Positive numbers are placed to the right of zero. - Negative numbers are placed to the left of zero. On the number line, the position of each number can be easily identified, and it is commonly used to help understand other concepts like the absolute value and basic arithmetic operations. Understanding the number line helps clarify how numbers relate to each other.

Imagine standing on zero and taking steps to the left and right. This analogy helps grasp operations and concepts like negative and positive directions, making abstract mathematics more tangible.

Distance from Zero

Distance from zero is a crucial concept in understanding absolute value. No matter if the number is positive or negative, the distance from zero is always a non-negative number.

- For a number like \(-3\), its distance from zero is 3. - For \(+3\), the distance remains 3.This concept helps in understanding why the absolute value of both \(-3\) and \(+3\) is 3. The absolute value essentially ignores the direction and only considers how far the number is from zero, making it always positive or zero.

Positive Numbers

Positive numbers are all numbers greater than zero. They are represented to the right of zero on the number line. Understanding positive numbers is essential, especially in the context of absolute values.

- When applying the absolute value, both negative and positive numbers are transformed into their positive equivalents. - This is because the absolute value operation measures the distance from zero.When viewing operations or calculations involving absolute values, it's important to remember that any negative number becomes positive. Thus, both \(-5\) and 5 have an absolute value of 5.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. They are the fundamental operations that allow us to manipulate numbers.

In the exercise, after finding the absolute values of \(-5\) and \(-2\), we use the addition operation to find the total absolute distance from zero:

• The absolute value of \(-5\) is 5.
• The absolute value of \(-2\) is 2.
• Adding these gives 7: \(5 + 2 = 7\).

Understanding these basic operations is a stepping stone to more complex calculations and is essential for solving problems involving absolute values efficiently.