Problem 47 - Beginning and Intermediate Algebra

Question

Find each sum. $$ [-3+(-4)]+[5+(-6)] $$

Step-by-Step Solution

Answer

The sum is -8.

Step 1: Identify the Expression

Recognize that the given expression is [(-3) + (-4)] + [5 + (-6)]

Step 2: Solve the First Bracket

First, solve the expression inside the first bracket: [(-3) + (-4)] To add two negative numbers, add their absolute values and then assign a negative sign to the result. |-3| = 3 and |-4| = 4; hence 3 + 4 = 7 So, (-3) + (-4) = -7

Step 3: Solve the Second Bracket

Next, solve the expression inside the second bracket: [5 + (-6)] To add a positive number and a negative number, subtract the smaller absolute value from the larger absolute value and assign the sign of the larger. |_6| = 6 and |5| = 5; hence 6 - 5 = 1 So, 5 + (-6) = -1

Step 4: Combine the Results

Now, combine the results of the two brackets: (-7) + (-1) To add two negative numbers, add their absolute values and then assign a negative sign to the result. |7| = 7 and 1 So, 7 + 1 = 8 So, (-7) + (-1) = -8.

Key Concepts

absolute value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. Put simply, it's always a positive number or zero. For example, the absolute value of -4 is 4, written as \(|-4| = 4\). The absolute value helps us measure the size of a number without considering its sign.
Understanding absolute value is crucial in integer operations, especially when dealing with negative numbers. It simplifies complex expressions by stripping away the negative sign for a moment, allowing for easier addition or subtraction.
For instance, consider \((-3) + (-4)\). Here, we first find the absolute values: \(|-3| = 3\) and \(|-4| = 4\). Then, we add these absolute values to get 7 and reintroduce the negative sign, resulting in \(-7\). By mastering the concept of absolute value, you'll handle negative numbers with ease and confidence.

negative numbers

Negative numbers represent values less than zero. They are essential in understanding real-world scenarios such as debts or temperatures below freezing. When performing arithmetic operations with negative numbers, maintain awareness of their unique properties.
Adding two negative numbers means going further into the negative side of the number line. For example, \((-3) + (-4) = -7\). Similarly, subtracting a negative number is like adding a positive. For instance, \((-5) - (-2) = -5 + 2 = -3\).
Visualizing negative numbers on a number line or using absolute values can help simplify operations and improve understanding. For example, in the expression \([5 + (-6)]\), treat it as \(|6| - |5| = 1\) and assign the sign of the larger absolute value. Here, it becomes \(-1\) because 6 has the larger absolute value.

positive and negative integer operations

Understanding how to operate with positive and negative integers is crucial for mathematical proficiency. Let’s break down some common operations:

• Adding Positive and Negative Integers: When adding a positive and a negative integer, subtract the smaller absolute value from the larger one. Assign the sign of the larger absolute value to the result. For example, \([5 + (-6)]\) becomes \(-1\) because \(|6| > |5|\).
• Adding Two Negative Integers: When adding two negative integers, add their absolute values and place a negative sign in front of the result. For example, \((-3) + (-4)= -7\).
• Subtracting Integers: To subtract an integer, add its opposite. For example, \([5 - (-6)]\) is the same as \([5 + 6]\) which is \([11]\).

By following these simple guidelines, you can manage operations with positive and negative integers efficiently. Practice these fundamental operations to strengthen your overall math skills.