Problem 44 - Algebra 1

Question

Find the sum. $$ -19+(-6) $$

Step-by-Step Solution

Answer

-25

Step 1: Understanding negative numbers

A key idea is that the addition of negative numbers results in a number that is more negative. This is because a negative number indicates a deficit or a loss. When you add a deficit to a deficit, the total deficit increases. So, -19 added to -6 will give a result that is more than -19.

Step 2: Adding negative numbers

To add -19 and -6, ignore the negative signs for now and just add 19 and 6 together. This gives 25.

Step 3: Incorporate the negative sign

Recall that because both numbers being added are negative, the result must also be negative. So, the answer to -19 plus -6 is -25.

Key Concepts

Negative Numbers

Negative numbers are used to represent values less than zero. These are numbers with a minus sign (-) in front of them. Think of negative numbers as representing losses or deficits.

• For example, -19 could represent a loss of 19 units, maybe 19 dollars in debt.
• They appear on the left side of a number line when zero is the midpoint.

Understanding negative numbers is fundamental in math since they behave differently from positive numbers in operations such as addition, subtraction, multiplication, and division. When handling negative numbers, visualize them as steps backward. Hence, adding more negative numbers is like stepping further back into the negatives. This concept will help when performing operations involving them.

Addition

Addition is the process of combining quantities. With positive numbers, it means increasing a value. However, when adding negative numbers, it's like combining losses or going farther into deficit.

• For instance, adding -19 and -6 is akin to increasing a debt from 19 to 25 by adding another 6.

Remember that when dealing with two negative numbers in addition, you're not reducing the negativity. You are actually enhancing it, just as adding more weights to a scale tips it further down. In this context, consider the operation \( -19 + (-6) \).Ignore the negative signs first and simply add the absolute values: \( 19 + 6 = 25 \). Given that both original numbers were negative, the result must be negative: \( -25 \).

Integer Operations

Integers include whole numbers and their negatives, so operations with them need special attention to rules, especially when involving negatives.

• When adding integers, the sign of the numbers dictates the direction of the sum result along the number line.
• For two negative integers, as seen in the exercise, the result continues in the negative direction.

This example shows that when two negative numbers, like -19 and -6, are added together, they push the sum further into the negatives, resulting in -25. Understanding integer operations entails recognizing patterns and rules: - Adding two negative integers will result in a more negative integer. - If you mix positive and negative numbers, you'll find the difference and keep the sign of the larger absolute value. This grasp of integer operations is essential as it applies to complex mathematics and real-world situations where debts, losses, or gains need computation effectively.