Problem 52 - Prealgebra

Question

Translate each of the following and simplify the result. Subtract \(-7\) from the sum of 7 and \(-12\)

Step-by-Step Solution

Answer

The result is 2.

Step 1: Sum Calculation

First, calculate the sum of 7 and \(-12\). This means we add these two numbers together: \(7 + (-12)\). Evaluate this expression. Since adding a negative number is the same as subtracting its absolute value, we get \(7 - 12 = -5\).

Step 2: Subtraction Execution

Now, we need to subtract \(-7\) from the result obtained in Step 1, which was \(-5\). Subtracting a negative number is equivalent to adding its positive equivalent. Therefore, \(-5 - (-7) = -5 + 7\).

Step 3: Final Simplification

Simplify the expression \(-5 + 7\) by performing the addition. Since \(7\) is greater than \(5\), find the difference and retain the sign of the larger absolute value number: \(7 - 5 = 2\). Thus, \(-5 + 7 = 2\).

Key Concepts

Integer Operations

Integer operations are fundamental in prealgebra and involve basic arithmetic with whole numbers, which can be either positive or negative. These operations include addition, subtraction, multiplication, and division. Unlike adding or subtracting positive numbers, when negatives are involved, the rules slightly change, and understanding these alterations is crucial.

When performing operations with integers, it's important to remember:

• A positive number added to a negative number is the same as subtraction.
• Subtracting a negative number is equivalent to adding the same positive number.
• The product or quotient of two integers with the same sign is positive.
• The product or quotient of two integers with different signs is negative.

Mastering integer operations is essential for solving equations and simplifying expressions smoothly. It lays the groundwork for more advanced mathematics, ensuring that future calculations are accurate and logical.

Addition and Subtraction of Integers

Skills in addition and subtraction of integers are vital in prealgebra. Dealing with different signs can be tricky, but with practice, it becomes intuitive.

Here’s a breakdown of how to approach these tasks effectively:

• Adding Integers: If the numbers are of the same sign, simply add the absolute values and use the common sign. For example, if you add 3 and 5, they are both positive so the answer is 8. If you add -3 and -5, the answer is -8.
• Different Signs: When the numbers are of different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. For example, adding 7 and -12 gives -5 because the larger absolute value is 12.
• Subtracting Integers: To subtract an integer, add its opposite. For instance, changing a subtraction like 7 - (-3) to 7 + 3 simplifies the operation.

Consistent practice is the key to mastering addition and subtraction of integers. This competence makes solving equations and managing negative numbers much simpler.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making calculations more efficient and less error-prone. It is an essential skill in mathematics that helps students quickly grasp the broader equation rather than getting lost in complex details.

To simplify expressions, follow these steps:

• Perform Operations Inside Parentheses first, following the order of operations.
• Combine Like Terms, which are terms that have the same variable raised to the same power.
• Carry out operations on numbers, such as adding, subtracting, multiplying, and dividing integers, as seen in earlier sections.
• Always double-check your steps to ensure all operations are correctly and logically executed.

In our exercise, simplifying involves careful application of integer operations to remove unnecessary complexity, leading to a clear and straightforward answer. When expressions are simplified, they become easier to interpret, further calculations become more manageable, and the potential for error diminishes significantly.