Problem 60 - Elementary Algebra

Question

Find the sums for the the following problems. \(1811+(-935)\)

Step-by-Step Solution

Answer

Answer: 876

Step 1: Determine the difference

First, find the difference between the two numbers without considering their signs: \(Difference = 1811 - 935 = 876\)

Step 2: Determine the sign of the result

Since the problem involves adding a positive number with a negative number, the sign of the greater absolute value number will determine the final sign. In this case, 1811 has a greater absolute value, and it is positive, so the final result will be positive.

Step 3: Write the final answer

The final answer is the difference found in step 1 with the sign determined in step 2: \(1811 + (-935) = 876\)

Key Concepts

Subtracting Integers

When it comes to subtracting integers, understanding the rules is crucial for solving problems accurately. We treat the subtraction of integers as the addition of a positive and a negative number. For example, when subtracting 5 from 10, written as \(10 - 5\), it's the same as \(10 + (-5)\). The rule to remember here is that subtracting a positive number is like adding its negative counterpart.

Let's apply this to a real problem: if we want to subtract -3 from 7, we can reframe this as \(7 + (+3)\), shifting a negative subtraction to a positive addition. It's essential to notice that subtraction can sometimes result in an increase in value when removing a negative.

In our original exercise, the subtraction was implied through addition of a negative integer: \(1811+(-935)\). Here, this engaged the rule that adding a negative is the same as subtracting its positive value.

Absolute Value

The absolute value of a number is the distance it is from zero on the number line, regardless of direction. It is always a non-negative number. In symbols, the absolute value of \(x\) is written as \(|x|\).

For instance, the absolute value of -5 is 5, and the absolute value of 5 is also 5, concisely \(|-5| = 5\) and \(|5| = 5\). Why is this important? When dealing with adding or subtracting integers, the absolute value tells us which number has more 'weight'.

In our exercise, we compared the absolute values of 1811 and -935, which are 1811 and 935, respectively. Since 1811 has a higher absolute value than 935, the result of \(1811+(-935)\) will take the sign of 1811, which is positive, leading to the final positive answer of 876.

Elementary Algebra

Elementary algebra encompasses the fundamentals of algebra, including operations on integers, which is what our exercise focuses on. It serves as the foundation for more advanced studies in mathematics. In this context, algebra helps us understand how to combine, subtract, multiply, and divide numbers along with understanding the properties of these operations.

When adding numbers with different signs, knowing the absolute value helps us to follow the algorithm effectively: identify the sign of the larger absolute value, subtract the smaller from the larger, and then apply this sign to the result. This is basic algebra at work, which directs us towards the solution diligently.

Ultimately, algebra is not just about finding the right answer, but also about understanding the process to get there. In the given exercise, following these foundational algebraic rules ensured we correctly solved \(1811+(-935) = 876\).