Problem 38 - Elementary Algebra

Question

Find the sums. \(10+(-2)\)

Step-by-Step Solution

Answer

Answer: The sum of 10 and -2 is 8.

Step 1: Identify the numbers and their signs

We have two numbers: \(10\) (positive) and \((-2)\) (negative).

Step 2: Find the absolute values of the numbers

The absolute value of a number is its distance from 0 without considering its sign. The absolute value of \(10\) is \(|10|=10\). The absolute value of \((-2)\) is \(|-2|=2\).

Step 3: Find the difference between the absolute values

Now subtract the smaller absolute value from the larger absolute value: \(10-2=8\).

Step 4: Determine the sign

Since the larger number (\(10\)) is positive, the result will also be positive: \(+8\).

Step 5: Write the final answer

The sum of \(10\) and \((-2)\) is \(10+(-2)=+8\).

Key Concepts

Absolute Value

Understanding absolute value is crucial when it comes to solving addition problems, especially when negative numbers are involved. Essentially, the absolute value of a number is its distance from zero on a number line, regardless of direction. So, it's always non-negative.

For example, the absolute value of both \(10\) and \( -10 \) is \(10\). This is represented as \(\lvert 10 \rvert = 10\) and \(\lvert -10 \rvert = 10\). The concept tells us how 'large' a number is without regard to its positive or negative sign.

Positive and Negative Numbers

Numbers can either be positive, negative, or zero. Positive numbers are greater than zero and are often represented without a sign, though you might see a plus symbol \( (+)\) in front of them sometimes. They are to the right of zero on the number line.

Negative numbers, which have a minus symbol \( (-)\) in front of them, are less than zero and to the left of zero on the number line. When dealing with addition problems involving both positive and negative numbers, remember that a negative is like subtracting that number, and a positive is like adding it.

Elementary Algebra

Elementary algebra is the branch of math that introduces algebraic expressions and the use of variables. While you may not see variables in basic addition problems, the rules of elementary algebra still apply.

Commutative Property of Addition

This property states that numbers can be added in any order and the result will be the same, e.g., \(3+5=5+3\).

Associative Property of Addition

This implies that when three or more numbers are added, the grouping doesn't affect the sum, e.g., \( (2+3)+4=2+(3+4)\). Understanding these properties can help streamline the process of solving more complex addition problems.

Addition with Negative Numbers

When performing addition with negative numbers, it can be helpful to consider combining losses and gains. If you have a positive number (gain) and you add a negative number (loss) to it, you're essentially reducing the amount of your gain.

For instance, in the problem \(10+(-2)\), you are adding a loss of \(2\) to a gain of \(10\). You'd find the absolute values, \(\vert 10 \vert = 10\) and \(\vert -2 \vert = 2\), and then subtract the loss from the gain, which gives you \(10 - 2 = 8\). Since the original number was positive and larger, the result is also positive, leading to the final answer of \(8\).

This process illustrates how understanding the underlying concepts of addition with negative numbers simplifies problem-solving and builds a strong foundation for more advanced mathematics.