Problem 21 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$-10+3$$

Step-by-Step Solution

Answer

The result of \(-10 + 3\) is \(-7\).

Step 1: Understand the Problem

The problem requires us to combine \(-10\) and \(+3\) using the rule for addition of integers, specifically focusing on combining a negative and a positive number.

Step 2: Apply the Rule for Addition

When adding a negative number to a positive number, we find the difference between the absolute values and give the result the sign of the number with the larger absolute value. Here, the absolute values are 10 and 3, and the difference is 7.

Step 3: Determine the Sign

Since 10 is larger than 3, the result will take the sign of -10, making the result negative. Therefore, the result is \(-7\).

Key Concepts

Exploring Positive and Negative Numbers

Positive and negative numbers serve as the backbone for understanding integer arithmetic. Positive numbers, like +3, are numbers greater than zero. They appear on the right side of the zero on the number line. Negative numbers, such as -10, are less than zero and located on the left side of the zero on the number line.

These numbers capture concepts of gain and loss, or moving right and left along a number line. When performing operations on these numbers, it's crucial to consider their position relative to zero. This understanding will aid you in determining outcomes during addition or subtraction exercises.

In algebra, positive and negative numbers allow us to express a wide range of quantities and grasp the idea of opposites, making them essential for problem-solving in mathematics.

Understanding Absolute Value

Absolute value provides the magnitude or size of a number without considering its sign. In simpler terms, it indicates how far away a number is from zero on the number line, irrespective of whether it's positive or negative.

For example:

• The absolute value of \(-10\) is \(|-10| = 10\).
• The absolute value of \(+3\) is \(|+3| = 3\).

The concept of absolute value is crucial when adding integers. It helps identify the effective distance from zero for forming comparisons and verifying larger values without sign interference.

In our example, knowing the absolute values assists in determining which number has a greater influence in the sum. This is why we took the difference between the absolute values when adding \(-10\) and \(+3\), resulting in \(7\) as the raw value.

Addition Rules for Positive and Negative Numbers

When adding positive and negative numbers, the approach can differ based on their signs.

Here are some basic rules to follow:

• When both numbers have the same sign, simply add their absolute values and give the result their common sign. For example, \(+5 + +2 = +7\).
• When the numbers have different signs, find the difference between their absolute values and assign the result the sign of the number with the larger absolute value, exemplified by \(-10 + +3 = -7\).

These addition rules make it easy to navigate through exercises involving different signs, ensuring that the solution maintains logical consistency connected to the magnitude of each number. Following these guides will lead to accurate arithmetic outcomes between integers representing real-world concepts like profit and loss, or ascents and descents.