Problem 82 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. $$ 8-(-9+5) $$

Step-by-Step Solution

Answer

12

Step 1: Simplify the expression inside the parentheses

First, simplify the expression inside the parentheses. You have \(-9 + 5\).Calculate \(-9 + 5 = -4\). So the expression now is \(8 - (-4)\).

Step 2: Simplify the subtraction of a negative number

Subtracting a negative number is the same as adding its positive counterpart. So, \(8 - (-4)\) becomes \(8 + 4\).

Step 3: Perform the final addition

Now, add the numbers: \(8 + 4 = 12\). So the result of the operation is 12.

Key Concepts

parentheses simplification

When simplifying mathematical expressions, it's crucial to handle the innermost parentheses first. This is where you clean up and reduce the complexity within these brackets. For example, in the exercise given, we start with 8 - (-9 + 5). To tackle this, simplify the expression inside the parentheses first. Calculate -9 + 5. Remember, when adding a positive to a negative number, it's the same as subtracting the smaller absolute value from the larger absolute value. This gives us -4. So now, the expression simplifies to 8 - (-4). Small and simple steps can help avoid any confusion or mistakes.

addition and subtraction of integers

Understanding addition and subtraction of integers is foundational in math. When you add integers, consider their signs. For instance, adding two positive integers is straightforward: their combined value is simply their sum. However, adding a positive integer to a negative one involves a bit more thought.

Let's take the example from the exercise: -9 + 5 = -4. Here, you subtract the smaller number (5) from the larger absolute value (9) and keep the sign of the larger absolute value, resulting in -4.

Subtraction can be thought of as adding a negative number. So, 8 - (-4) is the same as 8 + 4. The negative signs cancel out, turning the operation into a straightforward addition. It’s a useful trick to simplify problems: look at subtraction as adding a negative.

This concept also demystifies the step where 8 - (-4) converts into 8 + 4. Understanding this switch helps in other complex problems as well.

negative numbers

Negative numbers can be tricky but are essential in many areas of math. By definition, negative numbers are those less than zero, representing values to the left of zero on the number line.

When working with negative numbers in operations like addition or subtraction, it's helpful to remember a few rules:

• Adding a negative number is like subtracting its absolute value: i.e., 5 + (-3) is the same as 5 - 3.
• Subtracting a negative number effectively adds the absolute value: 7 - (-2) turns into 7 + 2.
• Two negative numbers added together become more negative: (-4) + (-5) becomes -9.

Returning to our example, handling –4 within the subtraction operation transforms 8 – (-4) into 8 + 4. It normalizes the problem and makes things simpler. Always remember to turn multi-step problems into manageable and smaller tasks for easier solutions.