Problem 18 - Beginning and Intermediate Algebra

Question

Find each sum. $$ 3+(-7) $$

Step-by-Step Solution

Answer

The sum of 3 and -7 is -4.

Step 1: Identify the Numbers

Observe the two numbers that need to be added: 3 and -7.

Step 2: Understand the Operation

Since one number is positive and the other is negative, this operation will essentially be a subtraction.

Step 3: Subtract the Numbers

Subtract the smaller absolute value from the larger absolute value: 7 - 3 = 4. The absolute values are taken for subtraction.

Step 4: Determine the Sign

Since -7 has a larger absolute value than 3, the result will carry the sign of -7. Therefore, the result is -4.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction. This means it’s always non-negative. For instance, the absolute value of 3 is written as \(|3| = 3\) and for -7, it’s written as \(|-7| = 7\).

Absolute values are used to simplify operations involving positive and negative numbers by focusing on their magnitude.
So when solving \(3 + (-7)\), the absolute values of 3 and -7 help us determine the numerical steps without initially considering their signs.

Positive and Negative Numbers

Positive numbers are greater than zero, and negative numbers are less than zero.
For the problem \(3 + (-7)\), 3 is positive and -7 is negative.
When combining positive and negative numbers, think of it as moving in different directions on a number line: adding a negative is like moving left, while adding a positive is like moving right.

This concept turns the addition \(3 + (-7)\) into a form where we understand that the operation essentially becomes a subtraction due to the different signs:

• Addition of positive and negative numbers cancels out a portion of each value.
• The sign of the larger absolute value (here, -7) determines the sign of the result.

Subtraction Operation

Subtraction involves finding the difference between two numbers. When adding a positive and a negative number, similar to \(3 + (-7)\), you subtract the smaller absolute value from the larger absolute value.

• First, find the absolute values: \(3\) and \(7\).
• Then subtract: \(7 - 3 = 4\).

Since -7 is the larger number in magnitude and it’s negative, the result carries this sign, resulting in the final answer of \(-4\). This step-by-step process makes it easier to handle operations involving different signs.