Problem 46 - Prealgebra

Question

Add the following numbers left to right. $$-8+(-3)+(-4)+(-7)$$

Step-by-Step Solution

Answer

The sum is -22.

Step 1: Add the First Two Numbers

We need to add the first two numbers: \(-8 + (-3)\)When adding two negative numbers, we keep the negative sign and add their absolute values:\(8 + 3 = 11\).So, \(-8 + (-3) = -11\).

Step 2: Add the Result to the Next Number

Next, we add the result from Step 1 to the next number, continuing the pattern:\(-11 + (-4)\).Again, we'll keep the negative sign and add the absolute values:\(11 + 4 = 15\).Thus, \(-11 + (-4) = -15\).

Step 3: Add the Final Number

Finally, add the result from Step 2 to the last number in the sequence:\(-15 + (-7)\).The same rule applies: keep the negative sign and add the absolute values:\(15 + 7 = 22\).Therefore, \(-15 + (-7) = -22\).

Key Concepts

Understanding Absolute Value

The concept of absolute value is central when working with negative numbers and their addition. Absolute value is the distance a number is from zero on the number line, *ignoring the sign*. Whether we are dealing with a positive or a negative number, its absolute value is always positive.

Let's take examples to clarify:

• For the number -8, its absolute value is 8.
• Similarly, for -3, the absolute value is 3.
• The absolute value of a positive number, such as 4, is simply 4 itself.

The absolute value helps simplify the process of adding negative numbers by converting them into their positive counterparts temporarily. When you add two negative numbers, you *add their absolute values* and then apply a negative sign to the result. This makes handling negative integers much more straightforward.

Step-by-Step Problem Solving

Breaking down problems into manageable steps is essential in mathematical problem solving. Here, we approached the addition of several negative numbers by using a step-by-step method. This approach ensures clarity and reduces errors.

**First Step: Adding the First Two Numbers**
In the original exercise, the first two numbers, -8 and -3, were added together. To do this, their absolute values were calculated, 8 and 3 respectively. These were then added to give 11, and the result was made negative, resulting in -11.

**Second Step: Continuing the Pattern**
Next, -11 was added to -4. Using the same method, we calculated the absolute values, added them to get 15, and applied the negative sign, resulting in -15.

**Final Step: Completing the Addition**
In the last step, -15 was added to -7. Following the familiar pattern, absolute values were added to make 22, thus the result is -22. Tackling complex operations with clear, systematic steps can lead to rapid problem solving and builds confidence.

Mastering Integer Operations

Integer operations are foundational in math, involving the basic processes of addition, subtraction, multiplication, and division with whole numbers. When dealing with the addition of integers, it is important to understand how to handle both positive and negative numbers appropriately.

When adding negative numbers, as illustrated in the original exercise, it is crucial to:

• Determine the absolute values of the numbers.
• Add these absolute values as if dealing with positive numbers.
• Apply a negative sign to the sum, since both numbers are negative.

This straightforward process ensures accuracy in calculations involving negative numbers.

Understanding these operations broadens mathematical competence and enhances the ability to perform more advanced math tasks. Mastery of integer operations also aids in real-world problems, such as calculating debts or temperatures, where negative numbers frequently occur.