Problem 85 - Introductory and Intermediate Algebra for College Students 4th

Question

Without using a number line, describe how to add two numbers with different signs. Give an example.

Step-by-Step Solution

Answer

To add two numbers with different signs, subtract the smaller absolute value from the larger absolute value and assign the sign of the bigger number. For instance, -3 + 5 = 2 and -4 + 2 = -2.

Step 1: Understanding the Concept

When adding two numbers with different signs, think of the signs as directions. The positive sign (+) indicates right or forward, and the negative sign (-) indicates left or backward. This means that by adding a positive number, you move forward, and by adding a negative number, you move backward.

Step 2: Applying the Concept

If you have two numbers, for instance, -3 and +5, since these numbers have different signs, you subtract the smaller absolute value from the larger absolute value. In this case, subtract 3 (ignore the sign) from 5. The result is 2. The sign of the larger absolute value number dominates the result. Since 5 has a larger absolute value and it's positive, the result will be positive. So, -3 + 5 = 2.

Step 3: Applying to Another Example

Consider another example, -4 + 2. Here, absolute value of -4 is larger than absolute value of 2. When subtracting, 4 - 2 = 2. As the larger absolute value is negative, the result will also be negative, so -4 + 2 = -2.

Key Concepts

Understanding Absolute Value

Absolute value refers to the distance a number is from zero on the number line, regardless of direction. It is always a positive number or zero. For example, the absolute value of both 5 and -5 is 5.

• Absolute value of a positive number is the number itself. For instance, |3| = 3.
• Absolute value of a negative number is the number without the negative sign. For example, |-3| = 3.
• The absolute value of zero is zero, meaning |0| = 0.

Absolute value helps simplify problems involving addition and subtraction of integers with different signs. It allows us to determine which number has a larger influence on the result, by comparing the absolute values of the numbers involved.

Positive and Negative Numbers

Positive and negative numbers are foundational elements of mathematics. They represent opposite directions or values on a number line. In everyday contexts, you might see positive numbers when dealing with profits or temperatures above zero and negative numbers when seeing debts or temperatures below zero.

• Positive numbers are greater than zero and are typically written without a sign. For example, 7 is a positive number.
• Negative numbers are less than zero and are usually denoted with a minus sign. For instance, -7 denotes a negative number.
• Zero is neither positive nor negative and serves as a dividing point between positive and negative numbers.

Understanding both positive and negative numbers is key when performing arithmetic operations like addition, subtraction, multiplication, and division. By recognizing how these numbers interact, one can easily solve problems involving integers.

Integer Operations

Integer operations include the basic arithmetic activities of addition, subtraction, multiplication, and division applied to whole numbers. Understanding how to efficiently conduct these operations is essential for problem-solving in math.

Adding Integers

Adding two integers involves considering their signs. When both numbers have the same sign, you add their absolute values and keep the sign. When they have different signs, subtract the smaller absolute value from the larger one, and take the sign of the number with the larger absolute value.

Tips for Integer Operations

• When adding integers with the same sign, just add and keep that sign. E.g., (-2) + (-3) = -5.
• When adding integers with different signs, subtract the absolute values and keep the sign of the larger absolute value number. E.g., (-5) + 3 = -2.
• Subtraction can be thought of as adding the inverse. For example, 7 - (-2) is the same as 7 + 2 = 9.

By mastering integer operations, solving more complex problems becomes manageable, as these operations form the basis for more involved mathematical concepts.