Problem 33 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$4-5-6$$

Step-by-Step Solution

Answer

The simplified expression is -7.

Step 1: Change Subtractions to Addition

Start by rewriting the expression given, changing each subtraction into the addition of its opposite. The expression is \(4 - 5 - 6\). First, we change \(-5\) to \(+ (-5)\), resulting in \(4 + (-5)\). Next, change \(-6\) to \(+ (-6)\). Now we have: \[4 + (-5) + (-6)\]

Step 2: Add from Left to Right

Now, solve the expression by adding from left to right. Calculate:- First, \(4 + (-5)\): - \(4 + (-5) = -1\).- Next, add \(-6\) to the result: - \(-1 + (-6) = -7\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics that involves rewriting mathematical statements in the easiest or most efficient form. When you simplify expressions, you aim to combine like terms and carry out operations to make the expression easier to understand and solve. For example, when faced with an expression like \(4 - 5 - 6\), the goal is to make it as straightforward as possible.To begin simplifying, convert subtractions into additions. Subtraction can often make an expression harder to manipulate, so express it instead as an addition of a negative. In our example, the expression \(4 - 5 - 6\) becomes \(4 + (-5) + (-6)\). This transformation is crucial as it prepares the expression for straightforward addition.When you have your expression in an addable format, proceed by carrying out the operations from left to right, just like reading text. This is not only a standard mathematical practice but also helps in organizing calculations, minimizing errors, and promotes better understanding of number interactions.

Basic Arithmetic Operations

The basic arithmetic operations include addition, subtraction, multiplication, and division. Knowing when and how to use these operations is key in mathematics. In prealgebra, addition and subtraction are often put together due to their interconnected nature. Ignoring signs for a moment:

• Addition is joining two or more numbers to get a larger number.
• Subtraction means taking away part of a number, resulting in a smaller number.

For our exercise, simplifying starts by dealing with subtraction by converting it to addition. Once the operations are in a consistent format (such as adding), they become easier to handle. Moving from left to right:1. First, add \(4 + (-5)\) to get \(-1\).2. Then, find the sum of \(-1\) and \(-6\) to achieve \(-7\).Consistent use of operations ensures clarity and efficiency when simplifying or solving expressions.

Negative Numbers

Understanding negative numbers is essential when working with expressions, especially when simplifying them. A negative number represents a value that is less than zero and lies on the left side of the number line. Negative numbers are indicated by a minus sign (\(-\)) as we see in \(-5\) and \(-6\).When working with negative numbers, remember:

• Adding a negative is the same as subtracting its positive counterpart.
• Subtracting a negative number is equivalent to adding its positive version.
• The combination of negative numbers strengthens their effect, moving deeper into negatives.

In our expression \(4 + (-5) + (-6)\), adding the first negative leads to an answer less than \(4\), shifting left on the number line to \(-1\). By adding another negative, \(-6\), the result becomes \(-7\), further emphasizing influence of the negative signs. Grasping this concept is vital as it helps predict outcomes of expressions involving negative numbers.