Problem 63 - Prealgebra

Question

Find each difference. $$-45-(-9)$$

Step-by-Step Solution

Answer

The difference is -36.

Step 1: Understand the Expression

We start with the expression \(-45 - (-9)\). This means we need to subtract -9 from -45.

Step 2: Simplify the Double Negative

Multiply the double negative to simplify the expression. Remember that subtracting a negative is the same as adding a positive: \(-45 - (-9) = -45 + 9\).

Step 3: Perform the Addition

Now, solve the addition \(-45 + 9\). This is the same as moving 9 units to the right on the number line from -45. The result is \(-36\).

Key Concepts

Subtraction of Integers

Subtraction is a fundamental operation when dealing with integers. When you see a subtraction such as \(-45 - (-9)\), it's important to interpret it correctly. Generally, subtracting a negative number is like adding its positive equivalent.
This means that the operation, in essence, becomes an addition: turning \(-45 - (-9)\) into \(-45 + 9\). This occurs because two negative signs cancel each other.
Always remember that subtracting any number effectively adds the opposite of that number. Thus, in the realm of integers, subtraction can be reimagined as addition, which simplifies numerous problems.

Adding Integers

Adding integers involves combining numbers to find their total. The integers can be both positive and negative.
Consider the expression \(-45 + 9\). Here, you are combining a negative and a positive integer. This scenario might seem tricky, but it's straightforward when visualized on a number line.
When adding a positive number to a negative one, start at the position of the negative number and move rightward by the value of the positive number.

• If the positive number is larger than the absolute value of the negative number, the result is positive.
• If it is smaller, the result remains negative.

For example, with \(-45 + 9\), you end up at \(-36\) after moving 9 steps to the right.

Number Line

A number line is an excellent visual tool to understand integer operations. It helps in seeing the effects of adding or subtracting numbers in a linear format.
When dealing with \(-45 - (-9)\), you'd start at \(-45\) on the number line. The process of subtracting a negative means you move positively. Therefore, instead of going left, move 9 units to the right. This visual movement lands you on \(-36\).
Using a number line offers a concrete representation, making abstract integer operations more accessible. It clearly demonstrates how addition and subtraction impact a position on the line, enhancing comprehension for learners.