Problem 35 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-8+3-4$$

Step-by-Step Solution

Answer

The expression simplifies to -9.

Step 1: Convert Subtraction to Addition of the Opposite

Start by changing the subtraction operations into addition of the opposite. The expression is initially \(-8 + 3 - 4\). Change \(- 4\) to \(+ (-4)\). The expression becomes:\[-8 + 3 + (-4)\].

Step 2: Add from Left to Right

Now, perform the additions from left to right in the expression \(-8 + 3 + (-4)\).First, add \(-8\) and \(3\):\[-8 + 3 = -5\]Next, add \(-5\) and \(-4\):\[-5 + (-4) = -9\]

Step 3: Simplified Expression

After completing the addition, the simplified result of the original expression \(-8 + 3 - 4\) is:\(-9\).

Key Concepts

Adding Integers

When dealing with integers, addition is one of the fundamental operations. Integers include both positive and negative whole numbers, as well as zero. Understanding how to add these is crucial when simplifying expressions. To add integers:

• If both integers are positive, simply add them to get the sum. For example, adding 3 and 5 gives you 8.
• If both integers are negative, add their absolute values and then assign a negative sign to the result. For instance, adding -3 and -5 involves adding 3 and 5 to get 8, then adding the negative sign results in -8.
• To add a positive integer and a negative integer, find the difference between their absolute values and give the sign of the larger absolute value to the result. For example, adding -8 and 3 means taking the difference, which is 5, and because 8 has a larger absolute value and is negative, the result is -5.

By carefully considering the signs, you can ensure you always get the correct result when adding integers.

Subtraction as Addition

A powerful technique in simplifying expressions is turning subtraction into addition. Instead of simply memorizing different rules for subtraction, it's often easier to think of subtraction as adding the opposite. Here's how it works:

• For any subtraction expression like "a - b", you can rewrite it as "a + (-b)". Here, you're adding the opposite of the second number.
• This method is particularly useful for clarity because it allows you to focus solely on addition, unifying the operation across the expression.
• If you have a complex series of numbers with subtraction and addition, first convert subtractions into additions of their opposites. For example, an expression like -8 + 3 - 4 becomes -8 + 3 + (-4).

By converting subtraction to addition, you simplify your approach, making the process of evaluating expressions more streamlined and less error-prone.

Left-to-Right Addition

When dealing with multiple components in an expression, a systematic left-to-right approach simplifies the process. Here’s why and how it helps:

• This approach ensures operations are performed in a consistent and orderly fashion, reducing confusion.
• For an expression like -8 + 3 + (-4), begin with the first two terms, -8 and 3. Perform the addition: -8 + 3 equals -5.
• Then, take the result from the first addition and add it to the next term: -5 + (-4) equals -9.
• Continuing systematically from left to right helps prevent errors, especially in longer expressions where keeping track of operations can be challenging.

By performing additions orderly from left to right, you simplify the evaluation of expressions, ensuring accuracy and clarity at every step.