Problem 70 - Algebra 1: Concepts and Skills

Question

Find the sum. $$ -11+(-8) $$

Step-by-Step Solution

Answer

\(-19\)

Step 1: Identify numbers

The two numbers given in the problem are -11 and -8.

Step 2: Rewrite addition of negatives as subtraction

Instead of writing it as -11 + (-8), we rewrite it as, -11 - 8. The addition of negative numbers can be rewritten as subtraction.

Step 3: Perform Subtraction

Subtract 8 from -11. The result is -19.

Key Concepts

Negative Number Addition

When we talk about negative number addition, it's crucial to understand that we are combining values that represent positions to the left of zero on the number line. Adding negative numbers might seem counterintuitive at first because we're essentially adding quantities that have a 'taking away' effect.

Consider the problem \( -11+(-8) \). It's helpful to think of negative numbers in terms of financial debt. If you owe \(11 (which can be represented as \( -11 \) in mathematics), and you incur an additional debt of \)8, you're effectively 'adding' another debt. This can be imagined as going further into the red. So here, adding \( -8 \) to \( -11 \) means accumulating more debt, leading us to a larger negative number, \( -19 \).

This operation can be represented visually on a number line where each step to the left signifies an increase in the negative value. Therefore, starting from \( -11 \) and moving 8 steps further to the left lands us at \( -19 \) on the number line.

Subtracting Integers

The process of subtracting integers is another fundamental piece of understanding arithmetic with negatives. When we subtract a positive number from a negative one, like in the problem \( -11 - 8 \), it's akin to moving further to the left on the number line, past zero and into negative territory. This is because subtraction is the process of taking away, and when we 'take away' from a negative number, we end up with a larger negative number.

It might be easier to comprehend this if we alter our perspective on subtraction. Rather than seeing subtraction as merely 'taking away,' consider it as measuring the distance between two numbers on the number line. Subtracting a larger number from a smaller one will yield a negative result, indicating that you've traveled a certain 'distance' into the negative side of the number line. This 'distance' in our problem is \( -19 \) which means we are 19 places to the left of zero.

Arithmetic Operations

Mastering arithmetic operations, such as addition and subtraction, is key to building a solid foundation in mathematics. These operations are the building blocks that lead to more complex mathematical concepts. Arithmetic with negative numbers is particularly important as it extends our ability to represent and calculate quantities that fall below zero, like debts or temperature drops.

Whether you're adding or subtracting negative or positive integers, the rules of arithmetic guide us to the correct answer. Remember that when adding two negative numbers, the result is negative, and it is numerically larger than the original numbers (e.g., \( -11 + (-8) = -19 \)). Conversely, if a positive and a negative number are combined, the result depends on which has the greater absolute value; for example, \( -11 + 8 \) would yield \( -3 \) since the negative number has the greater absolute value in this case.

Understanding these rules is crucial for working with real-life scenarios that involve calculation of gains and losses, temperature changes, or elevation differences, among various other applications.