Problem 50 - Prealgebra

Question

Translate each of the following and simplify the result. Find the difference of \(-7\) and \(-3\)

Step-by-Step Solution

Answer

The difference of -7 and -3 is -4.

Step 1: Identify the Operation

The problem asks for the difference between two numbers, which indicates that we need to subtract one number from the other. Specifically, it asks for the difference of -7 and -3.

Step 2: Write the Expression

To find the difference, we'll write the mathematical expression: \(-7 - (-3)\). Subtracting a negative number is equivalent to adding its positive, because of the double negative rule.

Step 3: Simplify the Expression

Re-write the expression by applying the double negative rule: \(-7 - (-3) = -7 + 3\). Now, proceed with the addition: -7 plus 3 equals -4.

Step 4: Conclusion

Thus, the simplified result of the expression for the difference of -7 and -3 is: \(-4\).

Key Concepts

Double Negative Rule

The double negative rule is a handy shortcut in arithmetic, especially when dealing with integer operations. When you see two negative signs in succession, like in a subtraction problem

• \(-a - (-b)\)

the double negative rule tells you that two negatives make a positive. This can be expressed mathematically as:

• \(-a - (-b) = -a + b\).

This rule stems from the property that subtracting a negative number is the same as adding its positive counterpart. For example, when dealing with

• \(-7 - (-3)\)

we can apply the double negative rule to simplify it to

• \(-7 + 3\).

Thus, embracing this rule can significantly simplify expressions and calculations involving integers.

Basic Arithmetic

Basic arithmetic is the foundation of all mathematics and includes essential operations such as addition, subtraction, multiplication, and division. These operations are used to perform calculations and solve problems. Understanding each of these operations is crucial to mastering more complex mathematical concepts.
In the context of subtraction, a common task is to find the difference between numbers, which can be particularly interesting when integers are involved. For example, finding the difference of

• \(-7\)
• and \(-3\)

involves subtracting the two numbers, which requires understanding how to manipulate negative numbers. By acknowledging that subtracting a negative number is equal to adding its positive (thanks to the double negative rule), you simplify the task to basic addition

• \(-7 + 3 = -4\).

Thus, mastering the basics of arithmetic helps in smoothly navigating through more complex operations.

Integer Operations

Integer operations encompass the mathematical procedures used when working with whole numbers, both positive and negative. These operations include adding, subtracting, multiplying, and dividing integers. Working with integers often introduces unique rules compared to positive numbers only, especially for addition and subtraction.
Maintaining a methodical approach when dealing with integers is crucial. The key lies in understanding the sign rules:

• When adding two integers with the same sign, keep the sign and add their values together. Example: \(-5 + (-4) = -9\).
• When adding two integers with different signs, subtract the smaller absolute value from the larger one, and use the sign of the number with the larger absolute value.Example: \(-7 + 3 = -4\).

For subtraction, remember the double negative rule:

• Subtracting a negative is the same as adding a positive.

Master these operations, and you're set for tackling any integer-based problems with confidence!