Problem 20 - Intermediate Algebra

Question

Perform the operations. See Example 1 . $$ 2+(-6)+(-3) $$

Step-by-Step Solution

Answer

The result of the operation is \(-7\).

Step 1: Identify the Operation

The expression given is \( 2 + (-6) + (-3) \). This is an addition operation that involves both positive and negative numbers. We will add these integers step-by-step.

Step 2: Add the First Two Numbers

Start by adding the first two numbers: \( 2 + (-6) \). This simplifies to \( 2 - 6 \). When you subtract a larger number from a smaller number, you will get a negative result. Here, \( 2 - 6 = -4 \).

Step 3: Add the Third Number

Next, take the result from Step 2, which is \(-4\), and add \(-3\) to it: \(-4 + (-3)\). When adding two negative numbers, you add their absolute values and keep the negative sign. Thus, \(-4 + (-3) = -7\).

Key Concepts

Negative Numbers

Negative numbers are integers that are less than zero. They are represented with a minus sign \((-\)). Imagine a thermometer: temperatures below zero are considered negative. Similarly, on a number line, negative numbers appear to the left of zero. This concept can be tricky because subtracting and adding negative numbers can change the expected outcome compared to positive numbers. Here are a few tips to remember:

• Subtracting a negative is the same as adding a positive. For example, \(3 - (-2) = 3 + 2 = 5\).
• Adding two negative numbers together results in a more negative number because you're adding the absolute values and increasing the negative sign.
• When a negative number is added to a positive number, you need to compare their absolute values to determine the result's sign.

Negative numbers are essential in operations like integer addition because they influence the sum depending on whether they pair with positive numbers or other negative numbers.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. It's always a positive number or zero itself. The absolute value is denoted by two vertical lines around the number, like this: \(|-3| = 3\).Features of absolute value include:

• It transforms any negative number into a positive one by considering only its magnitude.
• The absolute value of a positive number is the number itself, e.g., \(|5| = 5\).
• Understanding absolute value is crucial when adding and subtracting integers, especially when dealing with negative numbers.

When you add two negative numbers, you consider their absolute values—this gives you the total magnitude of both numbers added together. Then, you apply the negative sign to indicate the direction on the number line.

Step-by-Step Solution

Working through an integer addition problem step-by-step helps you understand each part of the operation. For the expression \(2 + (-6) + (-3)\), you tackle each addition one at a time.Here's how you solve it:

• Identify the Operation: Recognize you're working with both positive and negative numbers.
• Add the First Two Numbers: Combine \(2\) and \(-6\) by treating the addition of a negative as a subtraction (//)), resulting in \(2 - 6 = -4\).
• Add the Third Number: With your result being \(-4\), add the next negative number, \(-3\). Since both numbers are negative, simply add their absolute values and keep the negative sign, resulting in \(-4 + (-3) = -7\).

Following a step-by-step process ensures accuracy and builds confidence in handling similar math problems, reinforcing the understanding of operations involving integers.