Problem 31 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(-14+(-19)\)

Step-by-Step Solution

Answer

-33

Step 1 - Understand the Problem

The problem asks to add two negative integers: -14 and -19.

Step 2 - Combine the Absolute Values

When adding two negative numbers, add their absolute values. The absolute value of -14 is 14, and the absolute value of -19 is 19. So, calculate 14 + 19.

Step 3 - Perform the Addition

Add the absolute values: 14 + 19 = 33.

Step 4 - Apply the Sign

Since both numbers being added are negative, the sum is also negative. Therefore, -14 + (-19) = -33.

Key Concepts

absolute value

Absolute value is a key concept when working with integers. It refers to the distance a number is from zero on a number line, regardless of its direction. For any number, whether negative or positive, the absolute value is always non-negative. In mathematical notation, the absolute value of a number \( x \) is denoted as \( |x| \). For example, the absolute value of \( -14 \) is \( | -14 | = 14 \), and the absolute value of \( 19 \) is \( | 19 | = 19 \). Absolute values are important when adding negative numbers, as they allow us to focus on the size of the numbers without worrying about their signs.

negative integers

Negative integers are numbers less than zero, denoted with a minus sign (-). They represent values below zero, often used to indicate loss, decrease, or temperatures below freezing. When working with negative integers, especially in addition and subtraction, we need to understand how their signs affect the results.
For instance, adding two negative integers, like in our exercise where we add \( -14 \) and \( -19 \), involves combining their absolute values first before applying a negative sign to the result.
Example: To add \( -14 \) and \( -19 \), first find their absolute values: 14 and 19. Add these values: 14 + 19 = 33. Since both original numbers were negative, the sum is also negative: \( -14 + (-19) = -33 \).

integer addition

Adding integers can seem tricky, but it becomes easier with practice and understanding the rules. The process varies slightly depending on whether the integers are positive, negative, or a mix of both.
Adding Two Positive Integers: Simply add their absolute values. Example: \( 7 + 5 = 12 \).
Adding Two Negative Integers: Add their absolute values and then apply a negative sign to the result. Example: \( -7 + (-5) \rightarrow |7| + |5| = 12 \rightarrow \text{apply negative sign} = -12 \).
Adding a Positive and a Negative Integer: Subtract the smaller absolute value from the larger absolute value, then apply the sign of the number with the larger absolute value. Example: \( 7 + (-5) \rightarrow 7 - 5 = 2 \) (since 7 has a larger absolute value and is positive, the result is positive).
By understanding and applying these rules, you can confidently handle any integer addition problem.