Problem 79 - Prealgebra and Introductory Algebra

Question

Determine whether each statement is always true, sometimes true, or never true. Assume that \(a\) and \(b\) are integers. If \(a<0\) and \(b<0,\) then \(a+b<0\)

Step-by-Step Solution

Answer

The statement 'If \(a<0\) and \(b<0,\) then \(a+b<0\)' is always true.

Step 1: Understanding The Question

The problem statement provides us with the condition that \(a\) and \(b\) are negative integers and we need to evaluate if their sum will always, sometimes or never be negative as well. We need to analyze this situation and sift through all possibilities.

Step 2: Examining The Given Statement

We are given both \(a\) and \(b\) are negative integers. When two negative numbers are added together, the result will also be negative. This is because both numbers are pointing in negative direction and adding a negative number implies shifting further in the negative direction.

Step 3: Making The Conclusion

Since the sum of two negative integers will always result in a negative integer, we can conclude that the statement 'If \(a<0\) and \(b<0,\) then \(a+b<0\)' is always true.

Key Concepts

Understanding Integer Properties

Integers are the set of whole numbers that include zero, positive numbers, and their negative counterparts. One fundamental property of integers is that they are closed under addition; this means that when any two integers are added together, the result will always be an integer.

Considering negative integers specifically, it's crucial to understand their place on the number line. Negative integers are located to the left of zero. As you move left, the value of the integers decreases. If we took two negative integers, such as \( -3 \) and \( -5 \) and placed them on a number line, their positions would reflect their magnitude relative to zero.

Combining negative integers does not take you towards zero; instead, it takes you further away from it in the negative direction. Thus, the sum of two negative integers will always yield a more negative number. Properly comprehending this behavior is crucial to mastering operations involving integers.

Adding Negative Numbers

When it comes to adding negative numbers, it can sometimes be counterintuitive since we often associate addition with 'increasing' the result. However, in the world of integers, adding negative numbers will result in a number that is more negative, as confirmed by the integer properties.

A useful method to visualize this is using the number line. For example, if we're adding \( -3 \) to \( -4 \) on a number line, we start at \( -4 \) and move three units to the left because we're adding a negative number, which lands us at \( -7 \) as a result.

\textbf{Why is this the case?} Adding a negative number is equivalent to subtracting its positive counterpart. Hence, \( -3 + -4 \) is the same as \( -3 - 4 \) which is \( -7 \). This clearly illustrates that the sum of two negative numbers is a larger negative number.

Navigating Inequalities with Integers

Inequalities help us understand the relative size of different integers. When dealing with negative integers, it's important to know that a more negative integer is actually 'smaller' than a less negative integer. For instance, \( -5 < -2 \) because \( -5 \) is farther away from zero.

Within the framework of this question, we have an inequality involving the sum of two integers. Here's a critical tip: Adding two negative integers becomes a test of understanding how inequalities behave with negative numbers.

Since each of the two integers \(a \) and \(b \) is less than zero, their sum also needs to be less than zero as each addition of negative value 'decreases' the overall value. Therefore, it establishes the conclusion that if \( a < 0 \) and \( b < 0, \) then \( a + b < 0 \) is indeed always true. It reinforces the property that the sum of negative numbers will only cement their place further in the negative realm.