Problem 51 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ -25 \text { and }-10 $$

Step-by-Step Solution

Answer

The distance is 15.

Step 1: Identify the Numbers on the Number Line

First, recognize the two numbers you need to find the distance between: - The numbers are \(-25\) and \(-10\). - Both numbers are negative and are located on the left side of the 0 on a number line.

Step 2: Determine the Direction on the Number Line

Since both numbers are negative, determine which number is larger:- \(-10\) is greater than \(-25\) because it is closer to 0. - The movement from \(-25\) to \(-10\) is towards 0 on the number line.

Step 3: Calculate the Distance Using Absolute Values

The distance between two numbers on a number line is found by calculating the absolute difference:- Find the absolute value of both numbers: | \(-25\) | = 25 and | \(-10\) | = 10.- Calculate the absolute difference: \( | -25 - (-10) | \).

Step 4: Simplify the Expression

Solve the expression:- First compute: \(-25 - (-10) = -25 + 10 = -15\).- Take the absolute value of \(-15\), which is \(15\).Thus, the distance is \(15\).

Key Concepts

Understanding the Number Line

The number line is a straight line where every point represents a real number. On the number line, numbers to the right are greater, while numbers to the left are smaller. Zero sits in the middle, with positive numbers on the right and negative numbers on the left.

• Negative Numbers: Located on the left side of zero, becoming smaller as you move further to the left.
• Positive Numbers: Placed to the right of zero and increase as you move further right.

Understanding the number line helps you visualize the concept of distance, especially when dealing with negative numbers. Thinking of numbers as points on this line makes it easier to grasp differences and distances between them.

Exploring Absolute Value

The concept of absolute value is crucial when finding distances on a number line. The absolute value of a number is its distance from zero without considering its direction, represented by vertical bars like this: \(|x|\).

• Positive Numbers: The absolute value is the number itself, such as |5| = 5.
• Negative Numbers: The absolute value changes the sign, for example, |-5| = 5.

By using absolute values, we focus on the magnitude of numbers, making it straightforward to calculate distances. For example, in the exercise, finding the absolute value of \(-25\) and \(-10\) simplifies the process of determining the distance.

Working with Negative Numbers

Negative numbers can be tricky, especially when calculating distances. A negative number is simply a number that is less than zero and located on the left side of the number line.

• Closer to Zero: A negative number closer to zero is larger, like \(-10\) is larger than \(-25\).

When dealing with negative numbers:

• Be mindful of their order when calculating differences—subtract smaller values from larger ones.
• Calculating the distance involves using absolute values to ignore signs, focusing only on the size of the numbers.

So, moving from \(-25\) to \(-10\) involves taking the difference using their absolute values, providing a clear understanding of their distance apart.