Problem 69 - Algebra 1: Concepts and Skills

Question

Find the sum. $$ -6+2 $$

Step-by-Step Solution

Answer

\(-4\)

Step 1: Perform Addition

Add -6 and 2. When adding a negative number to a positive number, essentially it's like subtracting absolute values. So, it's equivalent to subtracting 6 from 2 which results in -4.

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Key Concepts

Negative and Positive Numbers

When dealing with numbers, understanding the concept of negative and positive numbers is foundational. A positive number is a number greater than zero, found to the right on the number line, while a negative number is less than zero, positioned to the left on the number line. Imagine the number line as a horizontal line with zero in the center, positive numbers extending to the right, and negative numbers stretching to the left.

Here's an easy way to visualize it: if you earn money, your bank balance goes up, represented by positive numbers. Spend more than you have, and you owe money, which can be depicted by negative numbers. In our exercise \( -6+2 \), \( -6 \) is a negative number indicating a deficit or removal, whereas \( 2 \) is a positive number, signifying an addition or gain.

Performing Addition

Adding integers, which include both positive and negative numbers, can be viewed as combining or removing quantities. If you have two groups you want to merge together, such as sheep and goats in a farm, you simply combine them to get the total count. Now, if some animals are leaving the farm (a negative contribution), you reduce that number from your total.

The operation can be straightforward when combining positive numbers, but adding a negative number changes the dynamic. Think of positive numbers as 'forward steps' and negative numbers as 'backward steps.' When we add a positive number, it's like taking steps forward; adding a negative number, on the other hand, is like taking steps backward. In our example \( -6+2 \), we're essentially taking 6 steps back and then 2 steps forward, ending up 4 steps backwards overall.

Subtracting Absolute Values

Absolute value represents a number's distance from zero on the number line, regardless of direction. It is always a positive number or zero, because distance cannot be negative. When adding a positive and a negative number, you can think of it as subtracting their absolute values to find the difference between them.

For instance, when subtracting \( 6 \) from \( 2 \), you consider how far apart the numbers are on the number line. Since the absolute value of \( -6 \) is \( 6 \) and the absolute value of \( 2 \) is \( 2 \), what you are really asking is 'how many steps does it take to get from 2 to -6?' The answer is 4 steps backward, hence \( -6+2 = -4 \). This method is useful because it allows us to understand and solve addition problems involving both negative and positive numbers effectively.