Problem 42 - Algebra A Combined Function

Question

Translate each phrase to an expression and simplify. Decrease 11 by -14 .

Step-by-Step Solution

Answer

The expression simplifies to 25.

Step 1: Identify the mathematical terms

The phrase "decrease 11 by -14" indicates that we need to subtract -14 from 11.

Step 2: Translate to a mathematical expression

The phrase can be translated to the expression: \[ 11 - (-14) \]

Step 3: Simplify the expression

Subtracting a negative number is the same as adding its positive. Therefore, the expression becomes:\[ 11 + 14 \]

Step 4: Calculate the result

Adding the numbers gives us:\[ 11 + 14 = 25 \]

Key Concepts

Translating Phrases to Expressions

In algebra, translating phrases into mathematical expressions is like decoding a sentence into a math problem. For example, when you hear the phrase "decrease 11 by -14", your first step is to identify what mathematical operation is needed. Here, "decrease" usually suggests subtraction. So, the phrase requires us to subtract \(-14\) from \(11\).
This translates into the expression \( 11 - (-14) \). Remember, the key is to understand words like "decrease", "more than", or "less than" to decide if you need to add, subtract, multiply, or divide. With practice, this process becomes quicker and simpler, leading to the correct algebraic expression that matches the phrase.

Simplifying Expressions

Simplifying expressions makes them easier to work with, helping you reach an answer more efficiently. In our example, the expression \( 11 - (-14) \) can be simplified by understanding what subtracting a negative number means. When you "subtract a negative", you're effectively adding the positive equivalent. This may sound confusing at first, but just think about changing the double negatives into a positive.
Thus, \( 11 - (-14) \) simplifies to \( 11 + 14 \). The simplification process is crucial for solving complex algebraic problems because it reduces errors and allows for straightforward calculations.

Addition and Subtraction of Integers

Working with integer addition and subtraction is a fundamental skill in algebra. Integers include positive numbers, negative numbers, and zero. The rules for adding and subtracting can guide you through any calculations. For instance, when adding integers like in our expression \( 11 + 14 \), you simply combine the values since they're both positive.
However, when subtracting, like \( a - b \), you're finding the difference. If \( b \) is negative, as in our case \( -(-14) \), subtraction becomes addition. Always remember these rules when dealing with integers:

• Add positives as usual.
• Add a negative by subtracting its absolute value.
• Subtract a negative by adding its absolute value.

Applying these principles makes integer calculations manageable and ensures you're using the right operations every time.