Problem 119 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. -14 and 22

Step-by-Step Solution

Answer

The distance between -14 and 22 on the number line is 36.

Step 1: Identify the Positions on the Number Line

First, recognize the two numbers you are working with: -14 and 22. On a number line, -14 is located to the left of zero, and 22 is located to the right of zero.

Step 2: Calculate the Absolute Difference

To find the distance between two points on a number line, calculate the absolute difference between the numbers. The distance is given by the formula \, |a - b| \, where a and b are the numbers. Here, we have:\[ |-14 - 22| \]

Step 3: Simplify the Expression

Simplify the expression inside the absolute value:\[-14 - 22 = -36\]

Step 4: Apply the Absolute Value

Take the absolute value of the result to find the distance:\[ |-36| = 36 \]

Key Concepts

Absolute Value: Understanding the Basics

Absolute value might sound complex, but it's quite simple! It measures how far a number is from zero on the number line, irrespective of which direction. Even if the number is negative, its absolute value is always positive because we're only interested in the magnitude, not the sign.

For instance:

• The absolute value of 3 is 3 because it's 3 units away from zero.
• The absolute value of -3 is also 3, as it too is 3 units from zero, just in the opposite direction.

Practically, think of absolute value as the "distance from zero." This concept helps in comparing distances and solving equations that require a non-negative outcome.

In our problem, calculating the absolute value of \(-36\) and obtaining 36 tells us the magnitude of the distance without regard to direction.

Distance Calculation on a Number Line

Calculating distance between two numbers on a number line is akin to measuring how far they are apart. The distance is independent of whether the values are negative or positive and can be found using absolute value.

To find the distance:

• Identify the two points, say `a = -14` and `b = 22`.
• Use the formula \( |a - b| \) or \( |b - a| \), both will yield the same result.

This simplifies using absolute values, ensuring the result is always positive, representing real-world distances.

So, whenever you see a math question asking for 'distance between' on a number line, you're simply tasked with finding the absolute difference, making this a straightforward task! In this case, \( |-14 - 22| = 36 \).

Elementary Algebra in Action

Elementary algebra introduces basic techniques useful across various math problems, including absolute value and calculation of distance. One such concept is simplifying expressions, a fundamental skill that helps solve more complex equations.

In our problem:

• First, understand the formula for distance: \( |a - b| \).
• Simplify the expression inside. For example, \-14 - 22 = -36\.
• Apply the absolute value to get \( |-36| = 36 \).

This step-by-step process leverages elementary algebra to solve real-world problems. By mastering these techniques, students can make quick work of distance-related questions.

Elementary algebra helps in understanding relationships and operations on numbers, laying a solid foundation for advanced mathematics.