Problem 104 - Algebra A Combined Function

Question

Each calculation below is incorrect. Find the error and correct it. See the Concept Check in this section. $$ -4+14 \stackrel{?}{=}-18 $$

Step-by-Step Solution

Answer

The correct result is 10.

Step 1: Identify the Arithmetic Operation

The given expression is \(-4 + 14\). This operation involves addition, even though there is a negative sign. We are adding \(14\) to a negative number \(-4\).

Step 2: Calculate the Sum

To solve \(-4 + 14\), start with the negative number \(-4\) and then add \(14\). This can be visualized as moving \(14\) steps to the right on a number line starting from \(-4\).

Step 3: Perform the Calculation

Starting at \(-4\) on the number line, moving \(14\) units to the right results in ending at \(10\). Therefore, \(-4 + 14 = 10\).

Step 4: State the Correct Result

The initial calculation \(-4 + 14 \stackrel{?}{=}-18\) is incorrect. Correctly, \(-4 + 14 = 10\).

Key Concepts

Understanding Negative Numbers

Negative numbers are essential in mathematics. They represent values less than zero. When thinking about negative numbers, envision them as positions to the left of zero on a number line. These numbers are designated with a minus sign (-) before them. For instance, - -4 is four units to the left of zero. - Sometimes, negative numbers can indicate loss or absence, like a debt or temperature below freezing. Understanding negative numbers is crucial when performing arithmetic operations, especially when combined with positive numbers. This knowledge helps us calculate differences and changes accurately.
Learning to handle negative numbers properly ensures we can solve problems involving them confidently.

Visualizing with the Number Line

A number line is an effective tool for understanding integer addition and subtraction, especially when dealing with negative numbers. Imagine a straight line with zero at the center. To its right, you'll find positive numbers (1, 2, 3, ...), and to its left, negative numbers (-1, -2, -3, ...). - This visual representation helps make sense of adding and subtracting. For the operation \(-4 + 14 \), you start at -4 on the number line. Then, move 14 units to the right. - Each step to the right involves adding +1. After moving 14 steps, you reach +10. Thus, a number line provides a visual method for tracking movement and accurately performing arithmetic operations involving both positive and negative numbers.

Mastering Arithmetic Operations: Addition and Subtraction

Arithmetic operations form the basis of much of mathematics, and understanding them with negative numbers is crucial. When performing addition with a negative number, interpret the negative sign as a direction. For example, in the expression \(-4 + 14 \), - you begin at -4. - Adding +14 means moving rightward on the number line.
When adding a positive and a negative number, the result depends on which number has the greater absolute value. - If the positive number is greater, as in \(-4 + 14\), the result is positive. In contrast, if the negative number is more significant, the result would be negative.
This understanding enables accurate calculations and corrects misconceptions like thinking \(-4 + 14 = -18\), when in fact \(-4 + 14 = 10\). Mastery of these operations is key to solving complex mathematical problems effectively.