Problem 32 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. Subtract 7 from the difference of -5 and 7 .

Step-by-Step Solution

Answer

The distance is 5.

Step 1: Identify the Numbers and Calculate the Difference

First, identify the given numbers on the number line: -5 and 7. To find the difference, calculate \(-5 - 7\). This yields -12. The difference of these two numbers on the number line is 12 because the distance is always positive.

Step 2: Subtract 7 from the Difference

Now, subtract 7 from the previously calculated difference of 12: \(12 - 7\).The result is 5.

Step 3: Conclude the Calculation

After calculating the above steps, we conclude that the solution to the exercise—the distance between the result of the subtraction operation and zero—is simply 5 as distances are always positive on a number line.

Key Concepts

Understanding Distance

When we talk about distance in mathematics, we are referring to the non-negative measure of how far apart two points are on a number line. It is important to remember that distance is always positive. This is because we are measuring the gap between two points without considering the direction.
For example, the distance between -5 and 7 is the same as between 7 and -5. We can calculate it by finding the absolute value of their difference. If you imagine a straight number line, distance helps us understand how far apart the numbers are, such as the space between floors in a building or between two positions on a map.

Subtracting Integers

Subtracting integers involves taking away one integer from another. This may seem straightforward, but it can sometimes trip people up especially with negative numbers. To subtract integers, we first need to identify the operation we are doing. For instance, if given the two numbers -5 and 7, and tasked with finding e.g., - (-5 - 7), we must remember: - Subtracting a positive number involves going further left on a number line, effectively decreasing the number. - Subtracting a negative number means moving right, hence increasing the number. In our original exercise, we did -5 - 7, resulting in -12. This tells us how much we need to move on the number line. Regardless of the direction, the absolute value (or size) of this operation tells us the distance—which is 12.

Distance on a Number Line

Understanding distance on a number line is crucial for visualizing and solving many arithmetic problems. Draw a number line from left to right, marking the positions of numbers like -5 and 7 along it.
To determine the distance, calculate the absolute value of their difference. Starting from -5, count the steps to get to 7, which involves covering all numbers in between. This step-by-step counting reflects the concept of absolute value, i.e., - The absolute value of - (-5 - 7) - Results in 12, representing no directional value. The final step of subtracting 7 from this distance simply moves further along the number line, once again emphasizing that distance ignores negative signs. This practical approach helps to measure real-world situations, like lengths or separations, without the complexity of sign handling.