Problem 44 - Algebra 2

Question

Find the value of each expression. $$ 8+(-5) $$

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Identify the Numbers

The expression given is \( 8 + (-5) \). Here, the numbers involved are 8 and \(-5\).

Step 2: Understand the Mathematical Operation

The operation in the expression is addition. When adding a negative number, it is equivalent to subtracting its absolute value from the other number.

Step 3: Convert Addition to Subtraction

Convert the expression from addition of a negative number to subtraction. Thus, \( 8 + (-5) \) becomes \( 8 - 5 \).

Step 4: Perform the Calculation

Subtract 5 from 8. This is a straightforward subtraction: \( 8 - 5 = 3 \).

Step 5: Verify Your Result

Double-check the calculation to ensure accuracy. Verify that \( 8 + (-5) \) simplifies correctly to \( 3 \), confirming our answer is correct.

Key Concepts

Addition of Negative Numbers

When you add a negative number to another number, it can feel a bit confusing at first. Essentially, adding a negative number is similar to subtracting the absolute value of that number. For instance, in the equation \( 8 + (-5) \), adding \(-5\) means moving 5 units to the left on a number line from 8. In other words, you are subtracting 5 from 8. This concept helps in understanding why \( 8 + (-5) \) can be rewritten as \( 8 - 5 \).

• Addition as Movement: Moving left means subtracting, and moving right means adding. So, when you add a negative, you're effectively moving in the opposite direction.
• Common Misunderstanding: It’s important not to get confused by the word "addition". Just think of it as a direction on the number line: negatives go left.

Subtraction

Subtraction is a fundamental operation in arithmetic where you take one quantity away from another. In our expression, \( 8 + (-5) \), we effectively perform the subtraction \( 8 - 5 \). The subtraction process is crucial because it helps in simplifying expressions that involve negative numbers.

• Simple Steps: Identify the larger number, subtract the smaller one, and retain the sign of the larger number if you're working with positives and negatives.
• Practical Example: From 8, remove 5: you have 3 left. It's that simple.
• Verification: Always take a moment to verify your solution. In our example, double-checking shows that taking away 5 from 8 indeed leaves you with 3.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering which direction it's in. This is always non-negative. For negative numbers, the absolute value is the same number but positive. In the expression \( 8 + (-5) \), the absolute value of \(-5\) is 5.

• Understanding Absolute Value: No matter if a number is positive or negative, absolute value discusses its magnitude or size.
• Usefulness: By focusing only on the size and ignoring the sign, absolute value helps convert problems to simpler subtraction.
• Visualizing on a Number Line: Imagine you are at zero, the absolute value tells you how far you need to step regardless of direction.