Problem 75 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(100[0.05(x+3)]\)

Step-by-Step Solution

Answer

5x + 15

Step 1: Distribute the Constant

First, distribute the constant 0.05 to both terms inside the parentheses. This gives us: \[ \begin{align*} 100[0.05(x) + 0.05(3)] & = 100[0.05x + 0.15] \end{align*} \]

Step 2: Multiply Each Term by 100

Next, multiply each term inside the brackets by 100. This gives us: \[ 100[0.05x + 0.15] = 100 \times 0.05x + 100 \times 0.15 \]Simplifying the products, we get: \[ 100 \times 0.05x = 5x \]\[ 100 \times 0.15 = 15 \]

Step 3: Combine the Terms

Finally, combine the terms to get the simplified expression: \[ 5x + 15 \]

Key Concepts

Distribution in Algebra

Distribution is a key concept in algebra that you use to simplify expressions. When you see an expression like \(a(b+c)\), you need to distribute the term outside the parentheses to each term inside.
This means multiplying \(a\) by both \(b\) and \(c\).
So, \(a(b+c)\) becomes \(ab + ac\).
Let's look at an example:
If we start with \(100[0.05(x+3)]\), the first step is to distribute 0.05 to \(x\) and 3:

\(0.05(x) + 0.05(3)\)

This gives us \(0.05x + 0.15\).
You just used the distribution property to break down a more complex expression into simpler parts.

Multiplying Constants

Once you have distributed terms in an expression, you often need to multiply constants to simplify further.
This process makes the numbers in your expression easier to manage and combine.
Let's take another look at our expression:
After distributing, we got \(0.05x + 0.15\).
Next, we'll multiply each term by 100:

\(100 \times 0.05x\) and \(100 \times 0.15\).

First, let's calculate \(100 \times 0.05\).

When you multiply 100 by 0.05, it's easier if you think about it in steps:
\(100 \times 0.05 = 100 \times \frac{5}{100} = 5\).
This gives us \(100 \times 0.05x = 5x\).
Now for \(100 \times 0.15\):

Multiply 100 by 0.15 in the same way:
\(100 \times 0.15 = 15\).
We now have \(5x + 15\) after multiplying the constants.

Combining Like Terms

The final step is to combine like terms.
Like terms are terms that have the same variable raised to the same power.
In our simplified expression \(5x + 15\), the \(5x\) is a term with the variable \(x\), and 15 is a constant term.

To combine like terms:

• First, identify the coefficients of the terms with the same variables. These are the numerical parts of the terms.
• Add these coefficients together if you’re combining addition type terms.

However, in our expression \(5x + 15\), there are no like terms to combine further, as they do not share the same variable.
Thus, \(5x + 15\) is the final simplified form.
Combining like terms makes expressions easier to interpret at a glance and is critical for solving more complex algebraic problems step by step.