Problem 40 - Beginning and Intermediate Algebra

Question

Perform the operation. See Example 3. Decrease 11 by \(-14\)

Step-by-Step Solution

Answer

The result is 25.

Step 1: Understand the Operation

The task asks you to decrease 11 by \(-14\). Decreasing by a negative number actually means addition. So, you need to add 11 and 14.

Step 2: Setup the Addition

Since decreasing by \(-14\) is the same as adding 14, the operation becomes: 11 + 14.

Step 3: Perform the Addition

Add the numbers: \[11 + 14 = 25\]

Step 4: Write the Result

The result of decreasing 11 by \(-14\) is 25.

Key Concepts

Addition

Addition is one of the basic operations in mathematics. It involves two or more numbers, called addends, which are combined to produce a total, known as the sum. The symbol for addition is the plus sign (+).

When you add numbers, you start from the rightmost digit, aligning the numbers vertically if they have more than one digit. You then add each column of digits, carrying over any value greater than 9 to the next column on the left. This process continues from right to left until all digits have been added, culminating in a final sum. Here’s a simple way to think about addition:

• Add each digit starting from the right.
• If a column sums to a value greater than 9, carry over to the next digit.
• Add these carry overs as part of the next column’s sum.
• End with the total sum of the numbers.

Understanding addition is fundamental to grasping more complex mathematical concepts like multiplication and algebra, as these operations often build upon the concept of addition.

Negative Numbers

Negative numbers are values less than zero, represented with a minus sign (-). They appear on the left side of zero on a number line.

When dealing with negative numbers, it’s essential to understand their unique rules in mathematical operations. In particular, when you "decrease" a number by a negative amount, you are actually **adding** the absolute value of that negative number. This is because subtracting a negative is the same as adding its positive counterpart. For instance:

• Subtracting -3 from 5 is the same as adding 3 to 5.
• Mathematically: \(5 - (-3) = 5 + 3\)
• Resulting in a final value of 8.

This concept ensures that mathematical operations with negative numbers are consistent and predictable. Understanding how negative numbers interact with other values is critical for performing calculations correctly, especially in algebra.

Basic Algebra

Basic Algebra uses symbols and numbers to express mathematical relationships succinctly. It involves variables which represent unknown values. Using operations like addition, subtraction, multiplication, and division, algebra allows us to create equations that model real-world scenarios or simplify complex problems.

In algebra, understanding the effect of each operation on the variable is crucial. For instance, if you see an equation like \(x - (-y)\), it can be simplified to \(x + y\). This relates back to the idea that subtracting a negative is equivalent to adding. Therefore, knowing basic algebra rules, such as:

• \(-a + b = b - a\)(tackling negatives in algebra)
• The distributive property: \( a(b + c) = ab + ac\)
• Balancing equations by performing the same operation on both sides.

These help simplify expression and solve equations efficiently. The basic principles of algebra make it a foundational element for higher-level mathematics and many scientific disciplines.