Problem 46 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -14+(-3)+11 $$

Step-by-Step Solution

Answer

The result is -6.

Step 1: Understand Addition and Subtraction with Negative Numbers

The given expression is \[-14 + (-3) + 11\]. Treat negative signs as instructions to move left or subtract on a number line, and positive signs to move right or add. We aim to find the overall value.

Step 2: Add the First Two Numbers

We start by adding the first two numbers: \[-14 + (-3)\].Adding a negative number means you are subtracting: \-14 - 3 = -17\.

Step 3: Add the Last Number

Now take the result from Step 2 and add 11:\[-17 + 11\].Move 11 steps to the right from -17, which brings you to \-6\.

Key Concepts

Addition and Subtraction

When dealing with addition and subtraction of negative numbers, it can initially seem perplexing, but it's quite logical once you break down the steps. Negative numbers are essentially numbers less than zero, positioned on the left side of a number line. When you're adding negative numbers, you actually move further to the left. For instance, adding \(-14\) and \(-3\), you're combining their distances to the left from zero.

In general, when you add a positive number, you move to the right and when you add a negative number, you move to the left. Similarly, subtracting is like doing the opposite: subtracting a negative number is akin to adding its positive counterpart because subtracting negative leads to movement to the right. Conversely, subtracting a positive number involves moving left along the number line. Understanding these principles aids in dealing with more complex integer arithmetic problems.

Number Line

A number line is a simple yet powerful tool to understand the position and operation of the numbers, especially when dealing with addition and subtraction of negative integers. Visualizing numbers on a line allows you to easily interpret movements to the left and right for arithmetic operations..

Let's illustrate this using the exercise \(-14 + (-3) + 11\). Initially, start at \(-14\) on the number line. Adding \(-3\) means moving three steps further to the left, reaching \(-17\). Next, adding \(11\) shifts your position eleven steps to the right, bringing you to \(-6\). This graphical depiction on the number line simplifies understanding of positive and negative movements, clarifying abstract numerical calculations.

• Negative addition equates to moving left.
• Positive addition equates to moving right.
• Use the number line to visually track these movements.

Understanding number line navigation builds a strong foundation for mastering integer arithmetic.

Integer Arithmetic

Integer arithmetic extends simple addition and subtraction to a more inclusive understanding of operations involving both positive and negative numbers. Integers are a set of whole numbers that include zero, positive numbers, and their negative counterparts. Mastering arithmetic with integers allows the solving of more complex expressions beyond basic operations.

In the original exercise, solving \(-14 + (-3) + 11\) required applying integer arithmetic rules. First, treating negative numbers involved recognizing that \(-3\) added to \(-14\) results in \(-17\), following the rule of moving left for subtraction-like addition. Adding \(11\) embodies integer arithmetic by moving right from \(-17\), showcasing how positive numbers influence the sum.

• Comprehending integer properties aids in simplifying and solving arithmetic problems.
• Engage in consistent practice to refine skills in integer operations.
• Explore different approaches to strengthening understanding and versatility in integer arithmetic.

By embedding these concepts, students can gain a firm grasp on handling various mathematical challenges confidently.