Problem 44 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-211-(-207)+3$$

Step-by-Step Solution

Answer

The simplified result is \(-1\).

Step 1: Convert Subtractions to Additions

We have the expression \(-211 - (-207) + 3\). The subtraction of a negative number is equivalent to the addition of the positive counterpart. Therefore, rewrite \(-211 - (-207)\) as \(-211 + 207\). Now, our expression becomes \(-211 + 207 + 3\).

Step 2: Add from Left to Right

Now, start by adding the first two numbers in the expression: \(-211 + 207\). Consider the absolute values: \(|-211| = 211\) and \(|207| = 207\). Since \(-211\) is greater in absolute value than \(207\) and is negative, the result will be negative: \(-4\).

Step 3: Finish Adding the Numbers

Now, add the remaining number, \(3\), to our result from the previous step: \(-4 + 3\). This is the sum of numbers with different signs, so calculate the difference between the absolute values: \(|-4| = 4\) and \(|3| = 3\). The result is the greater absolute value's sign: \(-1\).

Key Concepts

Addition of Integers

When we talk about adding integers, we're dealing with the combination of whole numbers that can be both positive and negative. It's essential to understand that the concept of adding integers is straightforward once you become familiar with handling their signs. Let's break it down:

• Adding two positive integers will always yield a positive result. Example: \( 2 + 3 = 5 \).
• Adding two negative integers gives a negative result. Example: \( -2 + (-3) = -5 \).
• When adding a positive and a negative integer, the larger absolute value determines the sign of the result. This means you find the difference between the two numbers, and the result takes the sign of the number with the greater absolute value. For instance, in the sum \( -211 + 207 \), since \( |-211| = 211 \) is larger than \( |207| \), the result is negative: \(-4\).

Being comfortable with these rules makes adding integers a breeze. Always remember that you're essentially finding the difference when the signs differ and applying the sign of the larger absolute value.

Opposites in Subtraction

Dealing with subtraction in math can be simplified by thinking about it as adding an opposite. What does this mean? Whenever you see a subtraction in an expression, you can rewrite it as the addition of the opposite value. Let's see how this works:

• If you have the expression \( a - b \), you can think of it as \( a + (-b) \). Now, let's apply this idea to subtraction involving negative numbers.
• The expression \( -211 - (-207) \) might look tricky at first, but by turning the second subtraction into the addition of the opposite, it becomes \( -211 + 207 \). This transformation simplifies our work since we're dealing with addition, which is often more intuitive.

This approach simplifies the operation significantly and can be a helpful tool across many mathematical problems. Try this method to make subtractions a little less daunting in your future calculations.

Absolute Value

The concept of absolute value is about distance. It measures how far a number is from zero on the number line, no matter in which direction. Let's dig deeper into understanding absolute values:

• The absolute value of a number, depicted as \(|a|\), is always non-negative. It strips the number of its sign.
• For instance, \|5\| is 5 because it’s 5 units away from zero. Similarly, though -5 is negative, \|-5\| is again 5 because it is also 5 units away from zero.
• When working with integers, knowing their absolute values helps in comparing their sizes regardless of their signs. Thus, in the expression \(-211 + 207\), we compared \| -211\| = 211 with \|207\| = 207 to determine that \(-211\) had a greater absolute value, leading to our final negative result, \(-4\).

Absolute values make it simpler to determine the magnitude of numbers when adding integers with different signs, allowing you to focus on the size without worrying about their direction on the number line.