Problem 52 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$14-3-(-7)$$

Step-by-Step Solution

Answer

The simplified expression is 18.

Step 1: Identify the first operation

In the given expression \(14-3-(-7)\), the first operation is \(14-3\). It's important to perform subtraction before addressing the negative number. Subtract 3 from 14 to get 11.

Step 2: Simplify the Negative

The next part of the expression is \(-(-7)\). To subtract a negative number it's the same as adding a positive number. Therefore, \(-(-7)\) simplifies to \(+7\). Replace \(-(-7)\) with \(+7\).

Step 3: Perform the Final Addition

Now you're left with \(11 + 7\). Add 11 and 7 to get 18.

Key Concepts

Understanding Addition and Subtraction

When simplifying algebraic expressions, addition and subtraction are foundational operations. They help you combine numbers to calculate a total or find a difference. In algebraic expressions, you may see a mix of these operations, like in the problem where you deal with both addition and subtraction in one line. To tackle these elements effectively:

• Identify the order of operations.
• Perform each operation from left to right.

For the given expression, start with subtraction because it appears first. Subtracting one number from another is just figuring out what remains when you take a part away. In this case, subtracting 3 from 14 (14 - 3) leaves you with 11. This step-by-step sequence is crucial to maintaining the integrity of your calculations as you move forward.

Working with Negative Numbers

Negative numbers can initially seem tricky, but they become simple with practice. They represent values less than zero and are indicated with a minus sign. Understanding how to handle these numbers is essential, especially in subtracting negative numbers.Consider the expression snippet \(-(-7)\). Here, you encounter a negative sign next to another negative number, which can be confusing at first.Here's a helpful trick:

• Subtracting a negative number is the same as adding its positive.
• Think of two negatives as a positive.

So in this exercise, subtracting \(-7\) from any number actually increases it by 7, not decreases. Thus, \(-(-7)\) becomes +7.When you get comfortable with this switch, handling negative numbers becomes much easier and straightforward.

Expression Simplification Techniques

Simplifying expressions is about making them as clear and concise as possible. This involves executing operations in the right sequence and removing any unnecessary complexity. For the expression given, start from left to right, just as you read a sentence.Here's how to simplify directly:

• First, perform simple calculations. For example, \(14 - 3\) becomes 11.
• Handle negative numbers next. Remember, two negatives make a positive, so \(-(-7)\) turns into +7.
• Finally, add up the simplified parts. 11 + 7 results in 18, which is the simplified result.

Expression simplification makes it quicker to interpret and use the mathematical results effectively. It also builds the foundation for tackling more complex algebraic problems efficiently.