Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$-1.5+(-5.3)$$

Step-by-Step Solution

Answer

-6.8

Step 1: Identify the numbers

The numbers to be added are -1.5 and -5.3.

Step 2: Follow the rules of adding negative numbers

When adding two negative numbers, the result will also be negative. So, the aim here is to add the absolute values of the two numbers and assign a negative sign to the result.

Step 3: Add the numbers

Adding the decimal numbers 1.5 and 5.3 (ignoring the sign for now) aligning their decimal point gives 6.8.

Step 4: Assign the negative sign

Because both numbers are negative, their sum will also be negative. Therefore, the result is -6.8

Key Concepts

Absolute Value

To clearly understand adding negative numbers, it's important to grasp what absolute value means. Absolute value is the distance of a number from zero on a number line, regardless of direction. It is always a positive number or zero. This concept is denoted by vertical bars, like \(|x|\). For instance, the absolute value of -1.5 is 1.5, and the absolute value of -5.3 is 5.3.

Understanding absolute value is crucial when dealing with negative numbers. When adding negative numbers, we often start by considering their absolute values to simplify the problem. When you focus on the absolute values and follow the rules, the calculations become easier.

Decimal Addition

Decimal addition requires aligning the numbers vertically by their decimal points before adding. Let's break down the sum -1.5 + (-5.3) by ignoring the negative signs initially.

• You align the decimals of 1.5 and 5.3 to easily perform the addition.
• Once aligned, you add digit by digit starting from the rightmost decimal place.
• Add 1.5 and 5.3 to get 6.8.

Even though the numbers are negative in our example, the process of decimal addition remains the same. Focus on the decimal values and adjust for the signs later in the calculation process.

Negative Sign

A negative sign indicates a value less than zero. It's important to handle these signs correctly in math operations. For addition, a negative sign means you move left on the number line. However, when adding two negative numbers, handling the negative sign correctly is crucial.

• The first step in such calculations is to ignore the negative sign and add the absolute values of the numbers.
• Once you have the sum of the absolute values, reapply the negative sign to the result.
• In the given example, adding the absolute values gives 6.8, which becomes -6.8 after assigning the negative sign back.

This process ensures the correct result when dealing with negative numbers in addition.

Rules of Integer Addition

The rules of integer addition are fundamental when working with negative numbers. These rules help you determine how to add integers regardless of their sign:

• Adding Two Positive Numbers: Simply add and keep the positive sign.
• Adding Two Negative Numbers: Add the absolute values and then assign a negative sign to the result.
• Adding a Positive and a Negative Number: Subtract the smaller absolute value from the larger absolute value, and the sign of the result is the same as the larger number.

For the calculation -1.5 + (-5.3), both numbers are negative, so the rule to add their absolute values and assign a negative sign gives the correct sum of -6.8. Following these straightforward rules can make adding different kinds of integers much easier.