Problem 107 - Algebra A Combined Function

Question

For Exercises 107 through 110 , determine whether each statement is true or false. The sum of two negative numbers is always a negative number.

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understand Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). For example, -2 and -3 are both negative numbers.

Step 2: Recognize the Rule for Adding Negatives

When you add two negative numbers, you sum their absolute values and retain the negative sign. For instance, adding -2 and -3 results in \[-2 + (-3) = - (2 + 3) = -5\].

Step 3: Determine the Sign of the Sum

Since both numbers being added are negative, the result will be a negative number, as shown in the example where -5 is still negative.

Step 4: Analyze Exceptions

Consider whether there is any scenario where adding two negative numbers does not yield a negative result. Since the sum of negative numbers combines their negative values, there are no exceptions.

Step 5: Draw Conclusion

Based on the analysis, the statement that the sum of two negative numbers is always a negative number is indeed true.

Key Concepts

Adding Negative Numbers

When it comes to adding negative numbers, the key is to understand what happens with their values.
Negative numbers are values less than zero. The process of adding them involves combining the sizes of these negative values and keeping the negative sign.
To visualize this, think about temperatures dropping or going deeper underwater. If you are at -2 degrees and drop another 3 degrees, you will be at -5 degrees.
This is because when you add -2 and -3, you are essentially adding their absolute values first.
So, it would look like this:

• Sum the absolute values: 2 + 3 = 5
• Attach the negative sign: -5

When you sum two negative numbers, the result is always a negative number. There's no scenario in standard arithmetic where this would change.

Absolute Values

To grasp the concept of absolute values, imagine removing the sign of a number.
The absolute value of a number is simply how far it is from zero, regardless of direction – just a positive measurement.

• The absolute value of -7 is 7.
• The absolute value of 3 is 3.

Absolute values simplify understanding how big a number is without considering its sign. For instance, when adding negative numbers, knowing their absolute values helps calculate their sum.
When adding -4 and -6, the absolute values 4 and 6 combine to be 10.
You then add the negative sign to get the final result: -10.
Thus, absolute values form the foundation for understanding the sum of negatives.

Rules for Adding Integers

Integer addition can be straightforward if you follow some simple rules:

• Adding two positive integers always gives a positive integer.
• Adding two negative integers always yields a negative integer, as their absolute values add and keep the negative sign.
• Adding a positive and a negative integer might be tricky: subtract the smaller absolute value from the larger one, and give the result the sign of the number with the larger absolute value.

This rule-based approach helps in determining the sign and magnitude of sums.
For example, adding -7 and 5 involves subtracting their absolute values

• |7| - |5| = 2

and keeping the sign of -7, resulting in -2.
Memorizing these rules makes managing and understanding integer addition easier.