Problem 24 - Elementary Algebra

Question

Find the sums. \((-3)+(-12)\)

Step-by-Step Solution

Answer

Answer: The sum of (-3) and (-12) is -15.

Step 1: Identify the numbers to be added.

In this problem, we have two negative numbers: \(-3\) and \(-12\).

Step 2: Find the absolute values of the numbers.

The absolute value of a number is its distance from zero on a number line, ignoring its sign. In this case, the absolute values of the numbers are: \(|-3| = 3\) and \(|-12| = 12\)

Step 3: Add the absolute values of the numbers.

Now, we add these absolute values together: \(3 + 12 = 15\)

Step 4: Assign the correct sign to the result.

Since we are adding two negative numbers, the result will also be negative. Therefore, the sum of \(-3\) and \(-12\) is \(-15\).

Step 5: Write down the final answer.

The sum of \((-3)+(-12)\) is equal to \(-15\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign (-). On a number line, negative numbers are located to the left of zero. When working with negative numbers, it's important to remember:

• They represent values below zero.
• Common examples include temperatures below freezing and losses in finance.
• When you add two negative numbers, the result becomes more negative.

For example, when you add a number like debt of \(-5\) and another of \(-10\), you end up with a larger debt, \(-15\). This is similar in principle to adding positive numbers but moves further away from zero.
Always use negative numbers to gain insights into scenarios where values are decreasing, like temperatures dropping or financial losses increasing.

Exploring Absolute Value

The absolute value of a number is its distance from zero on a number line, regardless of direction or sign.
This means it converts any number to a non-negative value, representing only the magnitude.
Here's what you need to know about absolute values:

• Symbolized by two vertical bars, like \(|-7| = 7\)
• Always non-negative.
• Useful for measuring distance or magnitude only.

In the context of our exercise, the absolute values of \(-3\) and \(-12\) are \(3\) and \(12\) respectively. Adding the absolute values helps us find the magnitude of change, while the sign of the original numbers helps assign the direction to the answer. The concept of absolute value simplifies complex problems by focusing on size without direction.

Basics of Integer Arithmetic

Integer arithmetic involves mathematical operations with whole numbers, both positive and negative, including zero.

Here are some key concepts for addition involving integers:

• When adding two positive integers, simply add their values.
• When adding negative integers, add their absolute values and keep the sign negative.
• If one is positive and the other negative, subtract the smaller value from the larger value in terms of absolute numbers, and keep the sign of the larger value.

In our exercise, we added \(-3\) and \(-12\). First, add the absolute values: \(3 + 12 = 15\). Since both numbers were negative, the result is \(-15\). This step-by-step understanding of integer arithmetic helps solve more complex problems and provides a clear pathway to the solution. Integer arithmetic is essential for dealing with everyday contexts where increases and decreases in values occur, like balancing a budget or calculating physical distances.