Problem 45 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-900+400-(-100)$$

Step-by-Step Solution

Answer

The simplified expression is -400.

Step 1: Rewrite the Expression

Start by changing all subtractions into additions by adding the opposite. The expression \[-900 + 400 - (-100)\]becomes:\[-900 + 400 + 100\]because subtracting a negative is the same as adding its positive.

Step 2: Perform the First Addition

Add the first two numbers from left to right. Start with the expression:\[-900 + 400\]Calculate the sum:\[-900 + 400 = -500\]

Step 3: Perform the Second Addition

Use the result from the previous step to add the next number:\[-500 + 100\]Calculate this value:\[-500 + 100 = -400\]

Key Concepts

Addition of Integers

Understanding the addition of integers is essential for simplifying complex expressions. Integers include both positive and negative numbers. When adding integers, one key rule is that a positive number and a negative number cancel each other out to a certain degree.

• Positive numbers represent a movement to the right on the number line.
• Negative numbers represent a movement to the left.

For example, if you add \[-900 + 400\] the operation means you start at \(-900\) on the number line and move \(400\) steps to the right, landing at \(-500\). Recognizing this movement helps make sense of the sum. Moreover, combining a negative with a positive involves comparing their absolute values and applying the sign of the larger value to the result. This method helps you understand the basic concept of adding integers.

Subtraction of Integers

Subtraction can be thought of as adding the opposite. This transformation simplifies expressions because it standardizes operations under a single rule. To subtract an integer, change the subtraction sign to an addition sign and switch the sign of the second number.

• For instance, subtracting \(-100\) from any number is the same as adding \(100\).
• Similarly, turning \(-800 - 50\) into \(-800 + (-50)\)

When you encounter a negative minus a negative, like \(-900 - (-100)\), rewrite it as \(-900 + 100\). Effortlessly transforming subtraction into addition allows for greater accuracy and understanding of integer operations.

Simplifying Expressions

Simplifying expressions is the art of reducing them to their most basic form. This process involves performing operations in the correct sequence. To begin, convert all subtractions to additions as described earlier. Here’s a simplified expression:

• Start with \(-900 + 400 - (-100)\). Change the expression to addition, resulting in \(-900 + 400 + 100\).
• Continue adding from left to right. Add \(-900 + 400\) first, which simplifies to \(-500\).
• Then, add the next number: \(-500 + 100\). This results in \(-400\).

Deal with each mathematical operation step by step. Be mindful of the integer signs and ensure conversions from subtraction to addition are correct. This thorough process leads to accurate simplifications and a deeper understanding of how integers interact.