Problem 55 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ -10 \text { and } 10 $$

Step-by-Step Solution

Answer

The distance is 20.

Step 1: Identify the Positions

Identify the positions of the given numbers on the number line. Here, we have two numbers: \(-10\) and \(10\).

Step 2: Calculate the Distance

The distance between two numbers on a number line is the absolute value of their difference. Calculate the difference between \(10\) and \(-10\): \[ 10 - (-10) = 10 + 10 = 20 \]

Step 3: Apply Absolute Value

Since distance is always a positive value, take the absolute value of the difference if necessary. Here, the difference is already positive, so the distance is simply \(20\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by two vertical lines, like this: \(|x|\). For example, the absolute value of both \(-5\) and \(5\) is \(5\). This is because distance cannot be negative; it is always positive or zero. When you calculate the distance between any two numbers on a number line, you use the absolute value of their difference.

• For instance, to find the distance between \(-10\) and \(10\), you first find their difference: \(10 - (-10)\).
• Then, take the absolute value of that difference, which is \(20\).

Absolute value helps ensure that no matter what numbers you choose, their distance is always expressed as a positive value.

Negative Numbers

Negative numbers might seem tricky at first, but they simply represent values less than zero. These numbers are placed to the left of zero on the number line. For example, \(-10\) is ten units to the left of zero. Recognizing and handling negative numbers is important when calculating distances between points on a number line.

• When you subtract a negative number, like when finding the distance between \(-10\) and \(10\), you are essentially adding its positive counterpart: that's why \(10 - (-10) = 10 + 10\).
• This mirrors the rule that subtracting a negative number is equivalent to adding its positive value.

Understanding how negative numbers work can make calculations on the number line easier and more intuitive.

Number Line Concepts

A number line is a visual representation that helps us understand numbers, their order, and their relationships to each other. It stretches endlessly in both the positive and negative directions, with zero in the middle. Each point on the line corresponds to a real number. A key concept is that the distance between two points on this line is the length of the path between them, measured in units.

• To calculate the distance on the number line between two numbers, like \(-10\) and \(10\), we use their absolute difference, ensuring it's always positive.
• This ensures that our understanding of distance remains consistent physically and mathematically.

Number lines are a fundamental tool in math that help visualize operations involving both negative and positive numbers, making concepts like distance, addition, and subtraction more tangible.