Problem 56 - Intermediate Algebra

Question

Simplify each numerical expression. $$ 6-9+11-8-7+14 $$

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Group Positive and Negative Numbers

Start by identifying and grouping the positive and negative numbers in the expression: Positive numbers: 6, 11, 14 Negative numbers: -9, -8, -7 Now we will add each group separately.

Step 2: Sum the Positive Numbers

Add all the positive numbers together:\[6 + 11 + 14 = 31\]

Step 3: Sum the Negative Numbers

Add all the negative numbers together:\[-9 + (-8) + (-7) = -24\]

Step 4: Combine the Results from Steps 2 and 3

Subtract the sum of negative numbers from the sum of positive numbers:\[31 - 24 = 7\]

Step 5: Final Solution: Simplified Expression Result

The simplified numerical expression is 7.

Key Concepts

Positive and Negative Numbers

In mathematics, positive and negative numbers are essential. Positive numbers are greater than zero and are usually presented without a sign or with a plus sign, like +5 or 8. Negative numbers are less than zero and are always shown with a minus sign, for instance, -3 or -7.
When working with math problems, it's crucial to recognize and handle these numbers correctly. This is particularly important when simplifying expressions, as misplacing a positive or negative sign can lead to incorrect results. Everyone should understand the rules that:

• Adding two positive numbers gives a positive result.
• Adding two negative numbers gives a negative result.
• When you subtract a negative number, it's the same as adding its positive counterpart.
• It's important to pay special attention when adding or subtracting them, especially if they appear together in a sequence.

Step-by-Step Solution

Breaking down mathematical problems into smaller, manageable parts makes it easier to solve them. This method, called a step-by-step solution, helps simplify complex problems, one part at a time. Let’s look at how this works with the exercise given.
First, identify the positive and negative numbers. Grouping them can simplify the process. In our exercise, numbers like 6, 11, and 14 are positive, while -9, -8, and -7 are negative. Next, handle each group individually:

• Positive Numbers: Add them together. For example, combine 6, 11, and 14, resulting in 31.
• Negative Numbers: Similarly, add these. Here, -9, -8, and -7 add up to -24.

Finally, combine the positive and negative totals: subtract the sum of negative numbers from the sum of positive numbers. This structured approach helps prevent mistakes, ensuring an easier path to the correct answer.

Addition and Subtraction

Addition and subtraction are fundamental operations in math, crucial for simplifying expressions and solving equations. Understanding how these operations work with positive and negative numbers is vital.

• Addition: When adding two positive numbers or two negative numbers, add their values and keep the original sign.
• Subtraction: To subtract a number, add its opposite. For example, subtracting -2 is the same as adding +2. This operation shifts numbers along the number line.

In the given exercise, begin by adding numbers within their sign groups, such as all positives and all negatives. Then, use subtraction to combine the sums from each group. This ensures accuracy in simplifying numerical expressions. Once you master these steps, you'll solve expressions more confidently and quickly.