Problem 25 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-11+(-5)$$

Step-by-Step Solution

Answer

The result of combining \(-11+(-5)\) is \(-16\).

Step 1: Understand the Problem

You need to combine two negative numbers: \(-11\) and \(-5\). When combining negative numbers, you perform regular addition on their absolute values and then apply the negative sign to the result.

Step 2: Add the Absolute Values

Calculate the sum of the absolute values of \(-11\) and \(-5\). The absolute values are \(11\) and \(5\). So, compute \(11 + 5 = 16\).

Step 3: Apply the Negative Sign

Finally, since we are combining two negative numbers, apply the negative sign to the result obtained in the previous step. Thus, the combined result is \(-16\).

Key Concepts

Understanding Negative Numbers

Negative numbers are essential in mathematics and represent values less than zero. You'll often see these numbers with a minus sign, such as

• \(-11\)
• \(-5\).

These numbers often represent losses or decreases.
For instance, a temperature of \(-5\) degrees indicates a point below freezing. Just like positive numbers have their rules, negative numbers follow specific mathematical properties when added or subtracted.
Understanding these properties is crucial for solving arithmetic problems involving them.

Decoding Absolute Values

Absolute values are an important concept when dealing with numbers, particularly negative ones. The absolute value of a number is its distance from zero on a number line, without considering its direction.
It is always a non-negative number, regardless of whether the original number is positive or negative.

• The absolute value of \(-11\) is \(11\) because it is eleven units away from zero.
• Similarly, the absolute value of \(-5\) is \(5\).

When adding or subtracting negative numbers, converting them to their absolute values can simplify the process.
This step helps in performing addition or other operations more efficiently.

Mastering Addition Rules

Addition rules help us consolidate multiple numbers into a single result.
For negative numbers, these rules slightly differ from those of positive numbers:

• When adding two negative numbers, add their absolute values together, just like regular numbers.
• After summing the absolute values, assign a negative sign to the result.
  • This step is crucial because it retains the original negative context of the numbers.

For example, when adding \(-11\) and \(-5\), we first calculate their absolute values: \(11 + 5 = 16\).
Applying the negative sign, as both were negative, our final answer becomes \(-16\).
These simplified steps help in managing more complex arithmetic operations involving integers.