Problem 155 - Prealgebra and Introductory Algebra

Question

Make up three subtraction problems such that each problem involves a negative number minus a negative number, and each problem has a difference of \(-8 .\) Then describe a strategy for writing these problems.

Step-by-Step Solution

Answer

The three subtraction problems with a difference of -8 would be: \[-1 - (-7) = -8,\] \[-3 - (-5) = -8,\] \[-2 - (-6) = -8.\] The strategy for creating these problems involves identifying pairs of numbers that sum to eight and applying a negative sign to each. Subtracting a negative number becomes adding a positive number, and the sum is subtracted from eight to yield a consistent difference of -8.

Step 1: Constructing the First Problem

Take two numbers that sum to 8. For instance, 1 and 7. Then, apply the negative sign to them. We now have -1 and -7. So, the subtraction problem will be -1 - (-7) which is equal to -1 + 7 = 6 - 8, yielding a difference of -8.

Step 2: Constructing the Second Problem

Take another set of two numbers that sum to 8. For instance, 3 and 5. Apply the negative sign to them so you now have -3 and -5. Thus, the subtraction problem would be -3 - (-5) = -3 + 5 = 2 - 8 = -6.

Step 3: Constructing the Third Problem

Take yet another set of two numbers that sum to 8. Let's say 2 and 6. Apply the negative sign to them to get -2 and -6. Hence, the subtraction problem will be -2 - (-6) = -2 + 6 = 4 - 8 = -4.

Step 4: Strategy for Writing These Problems

The key strategy for creating these problems is finding pairs of numbers that sum to 8 and then applying a negative sign to them. Subtraction of a negative number essentially becomes addition, and you end up subtracting the sum from 8 which consistently yields -8. The strategy utilizes the rule that subtracting a negative number is equivalent to adding a positive number.

Key Concepts

Negative Number Operations

Understanding how to operate with negative numbers is crucial in arithmetic and pops up frequently in various mathematical contexts. When you subtract a negative number, imagine the negative sign as an instruction to 'move in the opposite direction' on the number line.

For example, if you have the expression \( -5 - (-3) \) it is equivalent to \( -5 + 3 \). This is because subtracting a negative number is the same as adding its positive counterpart. The negative sign outside the parentheses changes the operation to addition, and the negative sign before the '3' is negated by the subtraction operation.

Therefore, the trick here is to remember that two negatives make a positive when it comes to subtraction. The problem simplifies to basic addition: \( -5 + 3 = -2 \), which means you are essentially moving '3 steps' in the positive direction starting from '-5' on the number line.

Subtraction Problem Solving

Subtraction with negative numbers can be simplified by following a consistent strategic approach. The solution involves the understanding that subtracting a negative is equivalent to adding a positive.

To illustrate, let's take the subtraction problem solving from our exercise: \( -1 - (-7) \). This problem can be rewritten as \( -1 + 7 = 6 \) using the rules of negative number operations. Following this, to find the difference that equals to \( -8 \), we need to adjust the equation to \( 6 - 8 = -2 \).

The strategy here can be broken down into a few digestible steps:

• Pair up two numbers that sum up to 8.
• Assign a negative sign to both numbers.
• Formulate a subtraction problem using these negative numbers.
• Rewrite the subtraction as addition (due to the double negative).
• Find the difference by subtracting 8 from the result.

This strategy ensures clarity and consistency when solving subtraction problems involving negative numbers, leading to the correct answer of \( -8 \).

Arithmetic with Integers

Integers include all the whole numbers, both positive and negative, as well as zero. Arithmetic with integers, therefore, deals with the addition, subtraction, multiplication, and division of numbers that can be less than, greater than, or equal to zero.

When subtracting integers, keep in mind the following rules:

• Subtracting a positive number moves the value to the left on the number line.
• Subtracting a negative number moves the value to the right on the number line (since it becomes addition).
• Adding a positive number moves the value to the right on the number line.
• Adding a negative number moves the value to the left on the number line (since it becomes subtraction).

These are the fundamental moves you make when performing arithmetic with integers. It's helpful to visualize these actions on a number line, which can guide your intuition when tackling a variety of integer problems.

Thus, effectively solving arithmetic problems with integers, such as the exercise provided, depends on one's grasp of these core principles and the ability to apply them correctly in calculations.