Problem 51 - Algebra for College Students

Question

Simplify each numerical expression. $$9-12-8+5-6$$

Step-by-Step Solution

Answer

The simplified expression is -12.

Step 1: Identify Operations

The numerical expression provided is \(9 - 12 - 8 + 5 - 6\). We need to simplify it by following the order of operations, which involves simply performing addition and subtraction from left to right as there are no parentheses or exponents to deal with.

Step 2: Subtract First Pair

Start by subtracting \(12\) from \(9\). \[9 - 12 = -3\]

Step 3: Continue Subtraction

Next, subtract \(8\) from the result \(-3\). \[-3 - 8 = -11\]

Step 4: Add the Next Number

Add \(5\) to the current result \(-11\). \[-11 + 5 = -6\]

Step 5: Final Subtraction

Finally, subtract \(6\) from \(-6\). \[-6 - 6 = -12\]

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying numerical expressions. Imagine you have several tasks to complete and want to do them in a logical sequence. In mathematics, the order of operations helps us determine this sequence for arithmetic calculations.

When an expression includes multiple operations (e.g., addition, subtraction, multiplication, division), you should perform them in a specific order to ensure the correct result. The commonly accepted guideline is PEMDAS:

• P: Parentheses - Always start evaluating the expressions within parentheses first.
• E: Exponents - Next, calculate the powers or square roots.
• MD: Multiplication and Division - From left to right, perform these operations as they appear.
• AS: Addition and Subtraction - Lastly, perform addition and subtraction from left to right.

In our given expression, since there are no parentheses or exponents, we focus on addition and subtraction, moving sequentially from left to right.

Addition and Subtraction

Addition and subtraction are fundamental arithmetic operations. Despite their simplicity, they can appear complicated in expressions with multiple terms, especially when dealing with negative numbers.

When you see an expression like the one given: - Start by identifying the operations. In our case, we're working with subtraction and addition. - Follow the order, performing each operation sequentially from left to right. In the expression \(9 - 12 - 8 + 5 - 6\), move from the left to the right:

• First, subtract \(12\) from \(9\), which gives you \(-3\).
• Then, take \(-3\) and subtract \(8\) to reach \(-11\).
• Next, add \(5\) to \(-11\), resulting in \(-6\).
• Finally, subtract \(6\) from \(-6\), yielding the final answer of \(-12\).

This step-by-step approach minimizes errors and miscalculations by maintaining a clear path through the problem.

Negative Numbers

Negative numbers often introduce a layer of complexity in arithmetic operations, but they are crucial for real-world calculations. Visualizing negative numbers can help to manage them more easily. Think of negative numbers as owing money or moving left on a number line.

In mathematical operations:

• Subtracting a positive number is like moving left on the number line. For example, \(9 - 12\) results in \(-3\); you move left past zero.
• Adding a positive number reverses direction, moving right. So, \(-11 + 5\) moves 5 steps right to \(-6\).
• Subtracting a positive number from a negative makes the negative larger (more negative). In \(-6 - 6\), you move further left, resulting in \(-12\).

When working with negatives, always keep track of the direction you're moving on the number line. This helps you maintain accuracy in computations and avoids common mistakes with signs.