Problem 42 - Beginning and Intermediate Algebra

Question

Explain why \(6-11\) is the same as \(6+(-11)\)

Step-by-Step Solution

Answer

Subtraction is adding the negative, so \(6 - 11\) is the same as \(6 + (-11)\).

Step 1: Understand subtraction in terms of addition

Subtraction can be thought of as adding a negative number. For instance, subtracting 11 from 6 can be rewritten as adding the opposite (negative) of 11 to 6. This is because subtraction of a number is the same as adding its negative counterpart.

Step 2: Convert subtraction to addition of negative

Convert the subtraction expression \(6 - 11\) into an addition expression by replacing the minus sign and the number subtracted with the negative of that number: \(6 - 11\) becomes \(6 + (-11)\).

Step 3: Verify equivalency using rules of arithmetic

When you add a negative number, you are effectively moving to the left on the number line. For \(6 + (-11)\), starting at 6 and moving left 11 places lands you at -5, which is the same result as calculating \(6 - 11\).

Key Concepts

Negative Numbers

Negative numbers are essential in everyday math. They are used to represent values less than zero.
Think of them like positions below sea level or temperatures below freezing.
In mathematics, positive numbers lie to the right of zero on the number line, while negative numbers lie to the left. Each negative number is the opposite of a positive number. For instance, the negative of 11 is -11, and this concept is vital when handling operations like subtraction.

Addition as Inverse Operation

Addition and subtraction are inverse operations.
Think of them as opposite actions. If you add a number to zero, and then subtract the same number, you return to where you started.

• Any subtraction can be converted to an addition problem by adding a negative.
• This approach simplifies complex calculations.

For example, instead of thinking about subtracting 11 directly, you can think of it as adding -11. Therefore, transforming an equation like \(6 - 11\)becomes\(6 + (-11)\).

Number Line

The number line is a visual tool used to represent numbers and basic arithmetic operations.
Each point on a number line corresponds to a number, with zero placed in the center.
On this line, moving right signifies increasing values, while moving left indicates decreasing values.

• Adding positive numbers moves you right.
• Adding negative numbers moves you left.

When you calculate\(6 + (-11)\),start from 6. Moving 11 units to the left lands you at -5. This makes the number line a powerful illustration of the relationship between operations.