Problem 50 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ -550-(-121) $$

Step-by-Step Solution

Answer

Answer: -429

Step 1: Analyze the expression

The expression given is: \(-550 - (-121)\). We can see that we have a negative number \(-550\) and we are subtracting another negative number \(-121\) from it.

Step 2: Subtracting negative numbers

When we subtract a negative number, it is the same as adding its positive counterpart. So, \(-550 - (-121)\) is equivalent to \(-550 + 121\).

Step 3: Perform the operation

Now, we simply need to add \(-550\) and \(121\). Since the numbers have different signs, we find the difference between their absolute values and assign the sign of the bigger number in the operation: Difference between the absolute values: \(|-550| - |121| = 550 - 121 = 429\) As \(-550\) is larger (in magnitude) than \(121\) and has a negative sign, the result will be negative: So, \(-550 + 121 = -429\) The final answer is \(-429\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero and are represented with a minus sign (-). For example, -550 behaves differently compared to positive numbers during mathematical operations. Understanding how they interact in various situations is crucial. In mathematics, negative numbers are used to represent values below zero, such as debts or temperatures below freezing. They follow specific rules when combined with other numbers. When two negative numbers are subtracted, like in our example

• The sign changes for the number being subtracted. So, in the expression \(-550 - (-121)\), the operation can be converted into an addition by changing the subtraction to addition and flipping the sign of -121 to +121.
• This implies that instead of subtracting a negative value, you add its absolute value.

Absolute Value

Absolute value refers to the distance a number has from zero on a number line, regardless of direction. It's always a positive quantity or zero. The absolute value essentially removes the sign of a number. It's represented using vertical bars, such as \(|-550|\) which equals 550. This becomes particularly important when comparing or performing operations with numbers of differing signs. Let's break it down in our context:

• For the number -550, the absolute value is 550.
• For 121, the absolute value is still 121 as it's already a positive number.
• When combining the absolute values in an operation like \(-550 + 121\), you determine the difference \(550 - 121\), which gives you 429.
• This difference helps in calculating the result of adding or subtracting integers with different signs.

Understanding absolute value helps in streamlining complex operations and ensuring that we process the numbers correctly.

Addition and Subtraction of Integers

Adding and subtracting integers involves rules that are crucial for accurate calculations. Remember, integers include both positive and negative numbers.There are a few straightforward rules when working with integers:

• When adding two positive integers, the result is straightforwardly positive (e.g., 4 + 3 = 7).
• When adding two negative integers, the sum is also negative, and you add their absolute values (e.g., -4 + (-3) = -7).
• When adding a positive and a negative integer, you subtract the smaller absolute value from the larger and take the sign of the larger absolute value (e.g., \(-550 + 121\) yields -429).
• Subtracting integers often involves adding the opposite. For instance, \(-(-121)\) becomes +121, changing the operation to addition.

It is vital to not only remember the rules but also know when to apply them, like converting subtraction into addition by reversing the sign of the integer being subtracted.