Problem 24 - Fundamentals of Mathematics

Question

For the following 18 problems, perform each subtraction. Use a calcula tor to cherk each result. $$ -6-8 $$

Step-by-Step Solution

Answer

The result of \(-6 - 8\) is \(-14\).

Step 1: Understand the Problem

The problem is asking us to perform a subtraction with two negative integers: \(-6 - 8\).

Step 2: Subtracting Negative Numbers

When subtracting integers like \(-6 - 8\), it is equivalent to adding the absolute value of the second number to the first, because subtracting a positive number means going left on the number line. So, this becomes \(-6 - (8) = -6 + (-8)\).

Step 3: Perform the Calculation

Calculate \(-6 + (-8)\) by adding the numbers as though they were positive and keeping the negative sign for the result: \[ -6 + (-8) = -14 \]

Step 4: Verify with a Calculator

Use a calculator to confirm the result. Input \(-6 - 8\) and check the displayed result, which should confirm that it equals \(-14\).

Key Concepts

Negative Integers

Negative integers are numbers less than zero. They usually have a minus sign in front, like this: -1, -2, -3. Negative numbers can cause confusion, especially when subtracting, because it's a bit different than dealing with positive numbers. Here are some key points about negative integers:

• They are found to the left of zero on the number line.
• Subtracting a negative integer is akin to adding its positive counterpart.
• Two negatives always "combine" to form a negative result (as in subtracting more than you actually have).

When solving problems involving negative integers, keep their position on the number line in mind. It helps to visualize the steps involved in subtraction.

Number Line

A number line is a useful tool for understanding integers and their relationships. It is a straight line with numbers placed at intervals; zero is typically at the center. The number line assists in visualizing operations like addition and subtraction.

• Positive numbers are to the right of zero, while negative ones are to the left.
• Moving left on the number line implies subtracting. If you go right, you're adding.
• It's useful for visualizing changes in value. For example, \(-6 - 8\) means starting at -6 and moving 8 steps to the left.

Understanding how to use the number line can clarify many seemingly difficult problems. It allows you to physically "see" the operation taking place.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It tells us how far a number is from zero without considering its sign. Absolute value is helpful when working with negative numbers.

• It is always a non-negative number.
• Denoted by vertical bars: \(|-8| = 8\), demonstrating that distance to zero is 8 units.
• When subtracting, knowing the absolute values can simplify the process: \(-6 - 8\) turns into \(-6 + (-8)\) because subtraction adds distances on the line.

Utilizing absolute value allows for a mental shortcut when completing subtraction tasks. It's like measuring the distance from zero without worrying about which side you are on.