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Chapter 1: Problem 76

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

Short Answer

Expert verified
6x + 30

Step by step solution

01

Distribute the 0.06

Distribute the 0.06 to both terms inside the parentheses: \[ 0.06(x + 5) = 0.06x + 0.3 \]
02

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: \[ 6x + 30 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檛 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.

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Most popular questions from this chapter

A manufacturer has fixed costs of \(\$ 1000\) to produce gizmos. Each gizmo costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the total cost to make \(x\) gizmos and \(y\) gadgets.

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. Work Exercises 139-142 in order, to find the average of \(23,18,13,-4,\) and -8 What is the average of the given group of numbers?

Use the distributive property to rewrite each expression. $$ -\frac{1}{4}(8 x+3) $$

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Write a numerical expression for each phrase, and simplify the expression. The sum of \(\frac{1}{2}\) and \(\frac{5}{8},\) times the difference of \(\frac{3}{5}\) and \(\frac{1}{3}\)
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