Problem 122 - Algebra A Combined Function

Question

Translate each phrase to an expression. Use \(x\) to represent "a number." The difference of a number and -10

Step-by-Step Solution

Answer

The expression is \(x + 10\).

Step 1: Identify Key Terms

The phrase given is 'The difference of a number and -10'. The key terms here are 'difference', which implies subtraction, and 'a number', which we will represent using the variable \(x\).

Step 2: Write the Expression

To express the difference in mathematical terms, we write the expression as \(x - (-10)\). Here, 'a number' is represented by \(x\) and we subtract -10 from \(x\).

Step 3: Simplify the Expression

When subtracting a negative number, the expression \(x - (-10)\) simplifies to \(x + 10\) because subtracting a negative is equivalent to adding the positive version of that number.

Key Concepts

Understanding the "Difference"

In algebra, understanding the term "difference" is crucial, as it directly translates to the concept of subtraction. When you encounter the word "difference" in a phrase, think of it as finding how much one quantity differs from another. This means you will be subtracting one value from another.

• The term highlights a comparison between two numbers.
• It typically involves a subtraction operation in mathematical expressions.
• For example, "the difference between 8 and 3" translates to the operation 8 - 3.

In the context of algebraic expressions, if we say "the difference of a number and -10," it means we are looking at how one unknown quantity, let's say the number represented by \(x\), differs from -10.

Grasping "Subtraction"

Subtraction is a fundamental arithmetic operation used to find the difference between numbers. In algebra, subtraction helps in translating relationships and expressions involving unknown values.

• When you subtract, you're essentially removing one quantity from another.
• The symbol for subtraction is a minus (-) sign.
• In expressions, subtraction can also help in rearranging terms and simplifying equations.

In this exercise, when we see the phrase "x minus -10," it can be confusing at first. But it’s important to remember that subtracting a negative number is like adding its positive equivalent. Hence, \(x - (-10)\) simplifies to \(x + 10\). The operation turns into an addition due to the negative sign being subtracted, which naturally switches its role to addition.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form. This often makes them easier to understand and solve. When simplifying, you'll look for terms that can be combined or operations that cancel each other out.

• Identify like terms and combine them if possible.
• Use arithmetic rules, such as subtracting a negative turning into an addition.
• Rearrange the expression for clarity and simplicity.

In the example provided, by simplifying \(x - (-10)\), we use the rule of subtracting a negative number. This simplifies the expression to \(x + 10\). Always look for these moments where a tricky subtraction expression can be made simpler by translating it into an addition.