Problem 10 - Beginning and Intermediate Algebra

Question

Subtract. See Examples 1 through 5 $$ -20-(-48) $$

Step-by-Step Solution

Answer

The result is 28.

Step 1: Identify the Operation

The problem asks you to subtract \(-48\) from \(-20\). This can be written as \(-20 - (-48)\).

Step 2: Rewrite Subtraction as Addition

When you subtract a negative number, it is equivalent to adding its positive counterpart. Rewrite \(-20 - (-48)\) as \(-20 + 48\). This stems from the rule that subtracting a negative is the same as adding the positive.

Step 3: Perform the Addition

Now, add the two numbers: \(-20 + 48\). Start with \(-20\) and simply add \(48\) to it.

Step 4: Calculation of the Result

Compute the result: \(-20 + 48 = 28\). To do this, think about starting at \(-20\) on the number line and moving 48 units to the right, which lands you at 28.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are often used to represent values below a defined zero point, like depths below sea level or temperatures below freezing. When you see a negative sign in front of a number, it indicates that the number is a negative number.

Negative numbers can seem tricky at first but are quite simple once you get used to them. They follow the same basic rules of arithmetic as positive numbers, but there are some special rules to keep in mind:

• Adding a negative number is the same as subtracting its positive counterpart.
• Subtracting a negative number is the same as adding its positive counterpart.
• Multiplying two negative numbers results in a positive number.

Understanding these basics will help you effectively tackle problems involving negative numbers.

Number Line

A number line is a visual representation of numbers placed in a straight line, usually from left to right, where each point corresponds to a number. It includes both positive numbers, negative numbers, and zero, acting as a central point.

Here are some key points about the number line:

• Moving right on the number line represents an increase, or addition, of numbers.
• Moving left signifies a decrease, or subtraction, of numbers.
• Zero is a midpoint that separates positive numbers from negative numbers.

Using a number line can be extremely helpful in visualizing operations, such as addition and subtraction of integers, and understanding how these numbers relate to each other. For instance, in our problem, starting at \(-20\) on the number line and moving 48 spaces to the right due to the rule of subtracting a negative being the same as adding a positive helps illustrate the result of 28.

Addition of Integers

The addition of integers is quite straightforward once you get the hang of using the number line. An integer can be a whole number, which might be positive, negative, or zero. When adding integers, you follow a few simple rules:

1. If both numbers are positive, the answer is positive.
2. If both numbers are negative, add them as normal numbers and add a negative sign to the result.
3. If one number is negative and the other is positive, subtract the smaller number from the larger (ignore the signs temporarily), then apply the sign of the larger absolute value.

• Example: Adding \(-20\) and \(48\) becomes a subtraction of absolute values: \(48 - 20 = 28\).

In the problem we looked at, the process was simplified by converting subtraction of a negative number into an addition operation, \(-20 + 48\), allowing us to use the easy integer addition rule and swiftly find the answer. Mastering these rules can provide a strong foundation for solving a wide array of math problems involving integers.