Problem 20 - Algebra 1: Concepts and Skills

Question

Find the difference. $$ -10-(-42) $$

Step-by-Step Solution

Answer

The difference is 32

Step 1: Understand the problem

In the given question we have -10 - (-42) to solve. Note that subtracting a negative number is equivalent to adding a positive number. This is because a negative number and a negative sign together becomes positive (The two negatives cancel each other out).

Step 2: Apply the rule

We apply the rule of two negatives becoming a positive, change -10 - (-42) to -10 + 42.

Step 3: Execute the operation

Upon executing the addition, we get 32, as -10 + 42 equals to 32.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, indicated by a minus sign (e.g., -1, -5, -10). They represent values below a neutral point, such as the freezing point on a thermometer.

Consider negative numbers as a step further left on the number line, contrasting with positive numbers that go right.

• Negative numbers act differently in operations compared to positive numbers.
• For example, subtracting a negative number is equivalent to adding its positive counterpart, turning two negatives into a positive.

A practical way to understand this is that when you remove a debt (negative), you gain money (positive), or when removing a loss, you gain a profit.

Addition and Subtraction Rules

Understanding the rules of addition and subtraction involving negative numbers can simplify calculations.

These rules help in transforming and solving problems more easily.

• When you subtract a negative number, it is equivalent to adding the positive version of that number.
• For instance, in our example problem: 1. Change \(-(-42)\) to \(+42\), so \(-10 - (-42)\) becomes \(-10 + 42\).

This transformation is guided by the specific rule: *Two negative signs become a positive*. This is a fundamental rule in algebra to simplify expressions with negative numbers.

Understanding these basic rules helps deal with more complicated algebraic expressions.

Integer Operations

Integer operations refer to computations involving whole numbers, both positive and negative. This includes operations like addition, subtraction, multiplication, and division.

A crucial aspect of integer operations is managing negative numbers effectively.

• The key to integer operations is consistency in applying operational rules, especially with negative numbers.
• For addition and subtraction, remember: - Adding a larger positive integer to a smaller negative integer results in a positive number (like \(-10 + 42\) in our exercise).- Adding a smaller positive integer to a larger negative may still result in a negative.

With clear understanding and practice of integer operations, handling negative and positive numbers becomes intuitive, aiding problem-solving and logical reasoning in algebra.