Problem 79 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ -58 \text { and }-34 $$

Step-by-Step Solution

Answer

The distance is 24.

Step 1: Identify the Absolute Values

First, find the absolute values of the given numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction.For \(-58\), the absolute value is \(58\).For \(-34\), the absolute value is \(34\).

Step 2: Determine the Distance Formula

To find the distance between two points on a number line, simply subtract the smaller number from the larger number and take the absolute value of the result. The formula is:\[|a - b|\]

Step 3: Apply the Formula to the Numbers

Use the formula to find the distance between -58 and -34:\[|-58 - (-34)| = |-58 + 34| = |-24|\]

Step 4: Compute the Result

Now, calculate the absolute value:\[|-24| = 24\]So, the distance between \(-58\) and \(-34\) on the number line is \(24\).

Key Concepts

Understanding Absolute Value

When we talk about absolute value, we're discussing how far a number is from zero on a number line. It doesn't matter whether the number is positive or negative. Absolute value is all about distance, so it’s always a positive number.
● Think of it as stripping off any minus sign in front of the number.
● For example, - If you have \(-5\), the absolute value is \(|-5| = 5\). - Likewise, for \(5\), the absolute value is also \(5\) because it is already positive.
In the context of our given numbers, the absolute value helps us work with distances easily, without the complication of handling negative values. This concept forms the foundation for many mathematical operations, including finding the distance on a number line.

Utilizing the Distance Formula

The distance formula on a number line is vital when you want to find out how far apart two numbers are. It essentially boils down to finding the absolute difference between the numbers. The formula is:\[|a - b|\]
This means you:

• Subtract the smaller number from the larger number to avoid a negative result.
• Take the absolute value of the result, ensuring a positive outcome that represents the distance.

For our exercise, this means we look at \(-58\) and \(-34\), calculate \((-58) - (-34) = -58 + 34\), yielding \(-24\).
Once you take the absolute value of \(-24\), it becomes \(24\), which is the distance between the numbers on a number line. This method allows for straightforward calculations no matter where the numbers are in relation to zero.

The Art of Subtracting Integers

Subtracting integers, especially when negatives are involved, can be tricky. However, there's a simple rule: subtracting a negative is the same as adding its positive counterpart. This concept makes it easier to work with problems involving subtraction of negative numbers.
For example:

• To solve \(a - (-b)\), you can rewrite it as \(a + b\).
• This transformation simplifies calculations significantly.

In our given problem, subtraction \((-58) - (-34)\) becomes an addition:\[ -58 + 34 = -24 \]
Once transformed, calculations with integers become more intuitive. By embracing this principle, subtracting integers becomes less daunting and more manageable, especially as the problems increase in complexity.