Problem 126 - Elementary and Intermediate Algebra

Question

Write \(-3-(-5)\) as addition of the opposite.

Step-by-Step Solution

Answer

The expression \(-3 - (-5)\) can be written as \(-3 + 5\). The result is 2.

Step 1: Identifying the Problem

Our task is to rewrite the expression \(-3 - (-5)\) as an addition problem. This will involve using the concept of 'addition of the opposite.'

Step 2: Understanding the Subtraction Operation

The given expression is \(-3 - (-5)\). The operation '-(-5)' means we are subtracting a negative number, which can be simplified by adding its opposite.

Step 3: Rewrite the Expression

To simplify \(-3 - (-5)\), we take the opposite of \(-5\), which is \(+5\). Therefore, the expression can be rewritten using addition: \(-3 + 5\).

Step 4: Perform the Addition

Now, perform the addition operation: starting from \(-3\), add \(+5\) to it. This can be visualized as moving 5 steps to the right on the number line from -3, which results in \(2\).

Key Concepts

Addition of the Opposite

When dealing with subtraction, a useful concept to simplify calculations is often referred to as "addition of the opposite." This approach involves changing a subtraction problem into an easier addition one. For the expression \[-3 - (-5)\], rather than subtracting the negative number \(-5\), you replace it with its positive counterpart, \(+5\). This transforms the problem into \(-3 + 5\). This method is powerful because it helps clarify operations and can make them more intuitive, especially when paired with number line visualization. Converting subtraction into addition of the opposite prevents common errors and streamlines arithmetic with negative numbers.

Number Line Visualization

A number line is a great tool to visualize integer operations, especially when dealing with negative numbers. Imagining numbers on a line can help you understand arithmetic operations as movements along that line. For instance, with \(-3 + 5\), imagine starting at \(-3\) on the number line. Adding \(+5\) means moving to the right, the direction of positive numbers. You would count 5 steps on the line: \(-2, -1, 0, 1, 2\). This shows how adding positive moves you right from the initial position. Using a number line helps make operations involving both negative and positive numbers more concrete, reducing possible confusion.

Subtracting Negative Numbers

Subtracting negative numbers can be tricky if you're not comfortable with them yet. The expression \(-3 - (-5)\) is an excellent example. When you see '-(-5)', think of how subtraction of a negative is like adding a positive number. The two negatives cancel each other out, essentially giving you the same result as (-3) + +5. Understanding this behavior is crucial because it's a common point of confusion. The trick is to remember that two negatives make a positive when subtracting, which turns subtraction into some form of addition. Take time to practice this as it will ease dealing with complex arithmetic problems in the future.