Problem 76 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(100[0.06(x+5)]\)

Step-by-Step Solution

Answer

6x + 30

Step 1: Distribute the 0.06

Distribute the 0.06 to both terms inside the parentheses: \[ 0.06(x + 5) = 0.06x + 0.3 \]

Step 2: Multiply by 100

Now, multiply each term by 100: \[ 100(0.06x + 0.3) = 100 \times 0.06x + 100 \times 0.3 \] Simplifying, we get: \[ 6x + 30 \]

Key Concepts

Distributive Property

The distributive property is a key concept in algebra. It tells us how to multiply a number outside the parentheses by each term inside the parentheses. Think of it like handing out candy to a group of friends. If you have a bag of candy and you want to give some to each friend, you distribute it equally.

Mathematically, this property is expressed as:
\[ a(b+c) = ab + ac \]
In our example, we started with \[100[0.06(x+5)]\]. We first distributed the 0.06 to both \(x\) and 5 inside the parentheses:
\[ 0.06(x+5) = 0.06x + 0.3 \].
This step helps break down the expression into simpler parts.

Multiplication

Multiplication is a basic arithmetic operation that involves adding a number to itself a certain number of times. In algebra, we often use multiplication to simplify expressions further.

After using the distributive property, our next step was to multiply each term by 100:
\[ 100(0.06x + 0.3) = 100 \times 0.06x + 100 \times 0.3 \].
This means you take each term and multiply it by 100 individually.

In this step, we got:
\[ 100 \times 0.06x = 6x \]
\[ 100 \times 0.3 = 30 \]
This simplifies our expression a lot, making it easier to understand and solve.

Combining Like Terms

Combining like terms is an essential skill in algebra. It simplifies expressions by adding or subtracting terms that have the same variable parts. It’s like grouping apples with apples and oranges with oranges.

In our example, the terms we have after multiplication are already combined since there are no like terms to add or subtract:
\[ 6x + 30 \].
This expression can’t be simplified further because 6x and 30 are not 'like terms'. 'Like terms' have the same variable and exponent, but here we have one term with an 'x' and one constant term.

Knowing when to stop is just as important as knowing how to combine terms. This final expression is our simplified version.