Problem 5 - Prealgebra

Question

Subtract. $$-3-5$$

Step-by-Step Solution

Answer

The result of \(-3 - 5\) is \(-8\).

Step 1: Identify the Operation

The problem requires us to subtract 5 from -3, which is written mathematically as \(-3 - 5\).

Step 2: Interpret Subtraction of Positive Number

Subtracting a positive number is equivalent to adding its negative. Therefore, \(-3 - 5\) is equal to \(-3 + (-5)\).

Step 3: Add the Numbers

Now we add the two negative numbers. When adding negative numbers, add their absolute values and keep the negative sign. So, \(-3 + (-5) = -(3 + 5) = -8\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers with a minus sign (-) in front of them, representing values less than zero. They often show up in situations like temperatures below freezing, debts, or elevations below sea level.

• In the number line, negative numbers are located to the left of zero.
• They are opposite to positive numbers.

When dealing with negative numbers in calculations:

• A negative plus a negative results in a more negative number.
• Subtracting a number is like adding its opposite (or negative).

It's important to handle negative signs carefully to avoid errors in calculations. For instance, in the operation \(-3 - 5\), we treat subtraction as adding a negative number. This reduces confusion and leads to the right answer.

The Role of Absolute Value

Absolute value is like a number's distance from zero on a number line, without regard to direction. This means it always results in a positive value. The symbol used for absolute value is two vertical lines, for example, \(|a|\).

• Absolute value ignores whether a number is negative or positive, focusing just on size.
• For example, \(|-3| = 3\) and \(|5| = 5\).

This concept is crucial when adding or subtracting integers. When adding negative numbers, calculate as if they were positive using their absolute values, then apply the negative sign to the result. For example, in our problem: \(-3 + (-5)\), calculate \(|-3| + |-5| = 3 + 5 = 8\), and then apply a negative sign giving \-8\.

Adding Integers Efficiently

Adding integers might seem easy, but it requires careful consideration of signs and absolute values. Here's how to handle them efficiently:

• **When both numbers are negative:** Add their absolute values and apply a negative sign to the sum, like in \(-3 + (-5) = -8\).
• **When one number is positive and one is negative:** Subtract their absolute values. The sign of the result takes the sign of the number with the larger absolute value.

For example, in the integer addition \(-3 + 5\), since \(5\) is larger, take \(5 - 3 = 2\) and the sign from \(+5\) means the result is \(2\).
One simple rule to remember is:

• Like signs mean add and keep the sign.
• Unlike signs mean subtract and take the sign of the larger absolute value.

Understanding these rules will help you confidently add any integers, regardless of their sign.