Problem 78 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ -34 \text { and }-23 $$

Step-by-Step Solution

Answer

The distance is 11.

Step 1: Understand the Problem

We need to find the distance between two numbers on a number line, which is essentially finding the absolute difference between those numbers.

Step 2: Determine the Absolute Difference

Calculate the absolute difference between the numbers \(-34\) and \(-23\). The absolute difference can be calculated using the formula \(|a - b|\) where \(a\) and \(b\) are the numbers. So, \(|-34 - (-23)|\).

Step 3: Simplify the Expression

Simplify the expression \(-34 - (-23)\). This is equal to \(-34 + 23\) because subtracting a negative is the same as adding the positive equivalent. Thus, the expression simplifies to \(-11\).

Step 4: Find the Absolute Value

Calculate the absolute value of \(-11\). The absolute value \(|-11|\) is \(11\), since absolute value is the non-negative value of a number.

Key Concepts

Absolute Difference

When finding the distance between two numbers on a number line, we refer to the absolute difference. Absolute difference is a mathematical concept that tells us how far apart two numbers are, without considering which number is larger or smaller. Think of it as the distance covered when moving between two points, regardless of direction. This is expressed mathematically as

• \(|a - b|\)

Here, \(a\) and \(b\) are the two numbers. In our case, the numbers are

• \(-34\) and \(-23\)

So you write it as

• \(|-34 - (-23)|\)

The subtraction of numbers in the absolute difference method gives you a positive measure of distance on the number line. Something to always remember here is:

• No matter how big or small the subtraction result is, taking its absolute value makes it positive and gives the distance.

Absolute Value

Absolute value is a fundamental concept in mathematics that pertains to the "magnitude" of a number. It's about how far a number is from zero on the number line, ignoring its sign. Simply put, it represents the "size" or "distance" of a number.For any real number \(x\), the absolute value is denoted as \(|x|\) and is defined as:

• \(|x| = x\) if \(x\) is positive or zero
• \(|x| = -x\) if \(x\) is negative

In our exercise, after simplifying the expression, we arrive at \(-11\). To find the absolute value, we take

• \(|-11|\)
• which results in \(11\)

This shows that

• \(-11\) is 11 units away from zero

Therefore, the absolute value provides us with a non-negative number, regardless of the original sign, which is exactly what we need for measuring distances.

Simplifying Expressions

Simplifying expressions is a core part of solving many math problems, especially when dealing with equations. The aim is to reduce expressions to their simplest form to make calculations easier and more intuitive.In our example, simplifying the expression \(-34 - (-23)\) involves a few straightforward steps:

• Transform the double negative into a positive, because subtracting a negative number is equivalent to addition:
  \(-34 - (-23) = -34 + 23\)
• Follow through with the arithmetic operation:
  \(-34 + 23 = -11\)

The process of simplifying helps to reveal the cleanest and most manageable form of the calculation, leaving you with the core value needed for further operations, such as finding the absolute value or making comparisons. Understanding this process is crucial in algebra and will support you in keeping complex problems under control.