Problem 26 - Elementary Algebra

Question

Find the sums. \((-4)+(-8)\)

Step-by-Step Solution

Answer

Answer: The sum of (-4) and (-8) is -12.

Step 1: Identify numbers to add

We are asked to find the sum of two numbers: \((-4)\) and \((-8)\).

Step 2: Apply the rule for adding negative numbers

When adding two negative numbers, we add their absolute values and keep the result negative. So, we have: \(|-4| + |-8| = 4 + 8 = 12\). Since both numbers are negative, our result will also be negative: \(-12\).

Step 3: Write the final answer

The sum of \((-4)\) and \((-8)\) is \(-12\).

Key Concepts

Absolute Value

Absolute value is the distance a number is from zero on a number line, without considering direction. It is always a non-negative number. For example, the absolute value of -4, written as \(|-4|\), is 4. Similarly, the absolute value of -8, \(|-8|\), is 8. Think of absolute value as "how far" away a number is from zero:

• It's like measuring the length of a string; regardless of the direction you measure it in, the length stays the same.
• The absolute value symbol, \(| \, |\), removes any negative sign from a number, turning it positive.

Understanding absolute value helps when adding and subtracting integers, especially when determining the magnitude of numbers without worrying about their signs.

Integer Addition

Adding integers, or whole numbers, can sometimes involve negative terms, which add a layer of complexity. When adding two negative numbers, you add their absolute values and keep the sign negative. Let's explore:

• Imagine two people with debts. If one owes \(4 and the other owes \)8, together they owe $12. Both debts add up because they are in the same direction (negative).
• For positive numbers, you simply add as usual, and the sign remains positive.

In our example, \((-4) + (-8)\), we used the rule of adding negative numbers: Add their absolute values to get \(|-4| + |-8| = 4 + 8 = 12\) and attach a negative sign, resulting in \-12\. The concept of integer addition helps you efficiently handle sums involving both positive and negative integers.

Number Line

A number line is a visual representation of numbers laid out in a straight line where integers are arranged in order from left to right. It is a powerful tool for understanding the relationships between numbers, especially with operations such as addition and subtraction. Here's how it helps with adding negative numbers:

• Numbers to the left of zero are negative, while those to the right are positive.
• Each step left or right represents a unit change in value.

When considering our example of \((-4) + (-8)\), start at -4 on the number line. You move 8 steps further left since you are adding a negative value. Ending at -12 helps visualize this process and shows why the sum of two negative numbers is larger in the negative direction. Understanding a number line boosts comprehension of integer operations by showing distance and direction visually.