Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$4+(-7)+(-5)$$

Step-by-Step Solution

Answer

The sum of \(4 + (-7) + (-5)\) is \(-8\).

Step 1: Identify Negative and Positive Integers

In the exercise, \(4\) is a positive integer, \(-7\) and \(-5\) are negative integers.

Step 2: Add Positive and Negative Integers

According to the rules of addition, adding a negative number is the same as subtracting a positive number. Therefore, the expression can be rewritten as \(4 - 7 - 5\).

Step 3: Perform the Subtraction Operations

Subtract \(7\) from \(4\), resulting in \(-3\). Then, subtract \(5\) from \(-3\), resulting in \(-8\).

Key Concepts

Positive and Negative Integers

Understanding positive and negative integers is essential as they form the building blocks of integer arithmetic. Think of positive integers as a step forward and negative integers as a step backward. When we talk about numbers like \(4\), \(7\), or \(5\) without any signs in front of them, they’re considered positive integers. They occupy the right side of zero on the number line.

Negative integers are the opposites; they’re represented with a minus sign \( - \) in front of them, like \( -7 \) and \( -5 \) in our exercise. These integers are located on the left side of zero on the number line. To quickly verify the sign of an integer, you can check if it comes with a minus \( - \) in front, which immediately indicates it’s a negative integer.

Integer Arithmetic

Integer arithmetic covers operations such as addition, subtraction, multiplication, and division involving both positive and negative numbers. One fundamental rule to remember is that two positive integers always result in a positive sum, whereas adding two negative integers will always result in a negative sum. It gets interesting when you mix the two.

Here’s where the concept of absolute values comes in handy. The absolute value of a number is its distance from zero on the number line, disregarding its sign. For instance, the absolute value of both \(4\) and \( -4 \) is \(4\). When adding or subtracting integers, comparing their absolute values can help determine the outcome’s sign and magnitude.

Subtracting Integers

Subtracting integers might seem tricky at first, but it becomes easier once you grasp the underlying concepts. Remember this key rule: subtracting a positive integer is the same as adding its negative counterpart. This means that \(4 - 7\) can be seen as \(4 + (-7)\). Similarly, when you subtract a negative integer, like in \( -3 - (-5)\), it’s the same as adding a positive one, resulting in \( -3 + 5\).

In the exercise \(4+(-7)+(-5)\), we follow the rule and rewrite \(4 - 7 - 5\) to find the sum systematically. We then subtract in steps: first \(4 - 7\), which is \( -3\), and then \( -3 - 5\), getting us to \( -8\). This step-by-step subtraction helps in solving the problem without confusion, and clearly reveals the final result as a negative integer.