Problem 11 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -9+(-3) $$

Step-by-Step Solution

Answer

The sum is -12.

Step 1: Identify the Operation

The problem involves adding two integers: -9 and -3. Since both numbers are negative, you will be adding their absolute values together.

Step 2: Add the Absolute Values

Compute the sum of the absolute values of -9 and -3. The absolute values are 9 and 3, respectively. Add these together: 9 + 3 = 12.

Step 3: Apply the Negative Sign

Since both original numbers were negative, the result of their sum should also be negative. Therefore, apply a negative sign to the result obtained in Step 2: -(12) = -12.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. To find the absolute value, you simply remove the negative sign if it exists. For example, the absolute value of \(-9\) is \(|-9| = 9\). Similarly, the absolute value of \(-3\) is \(|-3| = 3\). These values help to focus purely on the magnitude of a number.

• Absolute values turn negative numbers into positive equivalents.
• They indicate how large a number is, ignoring its sign.
• This concept is crucial in many mathematical operations, including absolute differences and distance calculations.

Understanding absolute value is essential when dealing with the addition of integers, especially when both numbers have the same sign.

Negative Numbers

Negative numbers are those less than zero and are typically represented with a minus sign. When you work with negative numbers, it's like moving in the opposite direction on a number line compared to positive numbers.

• Negative numbers decrease in value as their magnitude increases (e.g., -9 is less than -3).
• They are commonly used in real-life scenarios such as debt or temperatures below freezing.
• When adding negative numbers, you add their absolute values and keep the sign negative.

The operation with negative numbers often includes special rules, especially when they're combined with positive numbers or subtracted.

Zero as a Barrier

Between negatives and positives, zero acts as a separator. Moving from negative numbers toward positive numbers involves crossing zero, and vice versa. This is important to keep in mind when performing operations like addition or subtraction with mixed signs.

Addition of Integers

Adding integers can seem complex when negatives are involved, but it follows a simple set of rules. Let's understand how to handle different scenarios:

• **Same Signs**: When two numbers have the same sign (both are positive or both are negative), you add the absolute values and keep the sign. For instance, \(-9 + (-3) = -12\).
• **Different Signs**: If the numbers have different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value.
• **Zero**: Adding zero to an integer doesn't change its value. For example, \(5 + 0 = 5\).

These rules ensure consistency when you add integers, helping you to predict outcomes confidently.

Practical Tips

Always verify your solution using the context or number line. Consider real-world applications to better understand concepts. Integer addition is a fundamental part of mathematics, vital for solving problems efficiently.