Problem 117 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify: \(-8-2-(-5)+11 .\) (Section \(1.6,\) Example 3 )

Step-by-Step Solution

Answer

The simplified form of the expression \(-8-2-(-5)+11\) is \(6\).

Step 1: Identify Signs

Firstly, we have to identify the signs of the numbers. In our expression, \(-8\) is negative, \(-2\) is also negative, \( -(-5)\) becomes positive considering that two negatives make a positive when together, and \(+11\) is positive.

Step 2: Begin Computation

We start by adding the negative numbers together. So \(-8-2\) is equal to \(-10\). We then add the positive numbers together, so \(5+11\) is equal to \(16\). This leads to \(-10+16\).

Step 3: Final Computation

Finally, we do the final calculation by adding \(-10\) to \(16\), which gives us \(6\). Therefore, the simplified form of the given expression is \(6\).

Key Concepts

Algebraic Operations

Algebraic operations are the building blocks of algebra. They encompass the basic actions we perform on algebraic expressions, including addition, subtraction, multiplication, and division. In simplifying expressions like \( -8-2-(-5)+11 \), addition and subtraction are the primary operations at play.

One of the key strategies is grouping like terms: negative numbers with negative numbers and positive numbers with positive numbers. This allows us to manage the expression in parts, making it easier to visualize and solve. So, focusing on either all the negative terms or all the positive terms at one time can streamline the process.

By understanding how to properly combine terms through these operations, students develop the ability to simplify expressions efficiently and correctly. It's critical to keep track of the signs associated with each term, as they will affect the outcome of the operation.

Integer Arithmetic

Integer arithmetic involves the addition, subtraction, multiplication, and division of whole numbers, which can be either positive or negative. When simplifying expressions that contain integers, it is essential to follow the rules of arithmetic and always pay attention to the signs.

In our example \( -8-2-(-5)+11 \), combining the integers involves both the concept of adding and subtracting negative numbers. The effective use of integer arithmetic not only helps to obtain the correct answer, but it also builds a foundation for more complex math problems. Remember, subtracting a negative is the same as adding a positive, which is why \( -(-5) \) becomes a \( +5 \).

By mastering the fundamentals of integer arithmetic, students will be able to handle operations with negative and positive numbers more confidently.

Negative Numbers

Negative numbers, like \( -8 \) and \( -2 \) in our example, can sometimes be tricky for students. They are values less than zero, common in various mathematical concepts such as debts or temperatures below freezing.

Understanding how to work with negative numbers is crucial when simplifying algebraic expressions. The rules are straightforward: when you add a negative number, it’s like subtracting its absolute value. Conversely, subtracting a negative number is like adding its absolute value. Thus, \( -(-5) \) becomes a plus \( +5 \) because we’re subtracting a negative.

It’s helpful to visualize negative numbers on a number line or to think of examples like owing money (negative) versus receiving money (positive). This can make the concept more relatable and easier to understand, especially when simplifying expressions that are a mix of negative and positive terms.