Problem 33 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -9.6+(-3.5) $$

Step-by-Step Solution

Answer

The sum is \(-13.1\).

Step 1: Identify Numbers and Signs

We have two numbers: - The first number is \(-9.6\).- The second number is \(-3.5\).Both numbers have a negative sign in front of them.

Step 2: Consider Addition of Two Negative Numbers

When adding two negative numbers, the sum will also be negative. You add their absolute values and then place a negative sign in front of the result.

Step 3: Add Absolute Values

Calculate the absolute values first:\[ |-9.6| = 9.6 \]\[ |-3.5| = 3.5 \]Now add these absolute values:\[ 9.6 + 3.5 = 13.1 \]

Step 4: Apply Negative Sign to the Result

Since both numbers were negative, the result of their addition will be negative. Hence, the final answer is:\[ -(13.1) = -13.1 \]

Key Concepts

Absolute Value

The concept of "absolute value" is important when dealing with negative numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction. Essentially, it tells us how "big" the number is, without considering whether it is positive or negative.
For example:

• The absolute value of \(-9.6\) is \(9.6\).
• Similarly, the absolute value of \(-3.5\) is \(3.5\).

Think of absolute value as turning everything into a positive, non-negative form. This allows us to focus on magnitudes when performing operations. The symbols used to denote absolute value are two vertical bars, like \(|-9.6|\). Always remember, absolute values are never negative.

Negative Numbers

Negative numbers are numbers that are less than zero. On a number line, they appear to the left of zero. They represent things like debts or temperatures below freezing. Dealing with negative numbers can sometimes be tricky, especially in operations such as addition or subtraction.
Negative numbers are denoted by a minus sign in front of the number, as seen with \(-9.6\) and \(-3.5\) in this problem. When adding negative numbers, their absolute values might become relevant, as understanding their magnitude helps in calculating the total negative value.
Remember that:

• Add two negatives, and your result will also be negative.
• For example, \(-5 + (-6) = -11\).

This concept helps to understand why the result of the original problem is negative.

Arithmetic Operations

In arithmetic, operations such as addition, subtraction, multiplication, and division form the basic building blocks of mathematics. Addition involves combining two numbers to get their sum. However, the rules can vary depending on whether the numbers involved are positive or negative.
When adding two negative numbers, such as in this exercise, you take the absolute values of both, add them together, and then apply a negative sign to the result. Here's what happens step by step:

• Find the absolute values: \( |-9.6| = 9.6 \) and \( |-3.5| = 3.5 \).
• Add these values: \( 9.6 + 3.5 = 13.1 \).
• Lastly, apply the negative sign: \( -(13.1) = -13.1 \).

Understanding these steps can greatly simplify working with negative numbers in basic arithmetic operations.