Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$-5-(-19)$$

Step-by-Step Solution

Answer

The result of the expression \(-5-(-19)\) is \(14\).

Step 1: Identify the expression

The given expression is \(-5-(-19)\). Here, we are subtracting \(-19\) from \(-5\).

Step 2: Understand subtracting a negative

Mathematically, subtracting a negative number is equivalent to adding its positive counterpart. This is because \(-(-a)\) becomes \(+a\). So, \(-5-(-19)\) becomes \(-5+19\).

Step 3: Perform the addition

Now, sum the numbers \(-5\) and \(19\), which is equal to \(14\).

Key Concepts

Negative Numbers

Negative numbers can be a bit tricky at first, but understanding them is essential in mathematics. These are numbers that are less than zero, often represented with a minus sign (e.g., -5, -10). They can be thought of as the opposite of positive numbers.
Negative numbers are commonly used to represent losses, depths below sea level, or temperatures below freezing. They lie on the left side of a number line, which progresses from negative to positive values.
Understanding the nature of negative numbers is crucial when performing operations like addition and subtraction, particularly because the rules differ slightly from those involving only positive numbers.
To grasp negative numbers, consider:

• Number Line: A visual representation where negative numbers are to the left of zero.
• Opposites: For every positive number, there's a negative counterpart. For example, -3 is the opposite of 3.
• Contextual Use: Used to denote deficit or positions below a zero baseline.

Addition of Integers

Adding integers, whether they are positive or negative, involves a straightforward set of rules. We'll explore what happens when different types of integers are combined.
When adding two positive integers, the process is the same as adding natural numbers, resulting in a positive integer. For example, 3 + 7 equals 10.
When adding two negative integers, the numbers are treated like positive numbers, but the result retains the negative sign. For example, -3 + (-5) equals -8.
Most interesting is adding a positive integer and a negative integer, which requires finding the difference between their absolute values (ignoring the sign) and using the sign of the larger absolute value. For instance:

• If you have -5 + 3, calculate 5 - 3 to get 2, and since 5 is larger, the result is negative: -2.
• Conversely, for 3 + (-5), calculate 5 - 3 to also get 2, but since the positive number 3 is larger, the result is positive: 2.

A critical rule to remember is the principle of zero additions - adding zero to any integer does not change its value.

Subtraction of Integers

Subtraction of integers, particularly when negative numbers are involved, can often lead to confusion. But with some basic rules, it becomes manageable. The key to mastering subtraction with integers is to understand the concept of subtracting negatives.
Subtracting an integer is equivalent to adding its opposite. Here's how it works: instead of subtracting a number directly, you "add" the opposite value. So, if you have -5 - (-19), you transform this to -5 + 19.
Here's a general guideline:

• Subtract a Positive: Simply move to the left on the number line. E.g., 10 - 5 is the same as moving 5 spaces left from 10, resulting in 5.
• Subtract a Negative: Convert the subtraction into addition. This means moving right on the number line. E.g., 5 - (-3) is the same as 5 + 3, resulting in 8.

Understanding subtraction as adding the opposite helps simplify complex problems and solve them efficiently. Remember that the operation changes according to the second number's sign, and applying these rules helps maintain clarity.