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Chapter 4: Problem 61

Solve the equations and inequalities. $$0.035 x+0.052(x+5,000)=1,130$$

Short Answer

Expert verified
x = 10,000

Step by step solution

01

- Distribute

First, distribute the term inside the parentheses. Multiply 0.052 by both terms inside the parentheses: \[0.052(x + 5000) = 0.052x + 0.052 \times 5000\]. This simplifies to \[0.052x + 260\].
02

- Combine like terms

Now rewrite the equation as: \[0.035x + 0.052x + 260 = 1130\]. Next, combine like terms: \[0.087x + 260 = 1130\].
03

- Isolate the variable term

Subtract 260 from both sides to isolate the term with the variable: \[0.087x = 1130 - 260\]. This simplifies to \[0.087x = 870\].
04

- Solve for x

Divide both sides by 0.087 to solve for \(x\): \[x = \frac{870}{0.087}\]. Calculate the division: \[x = 10,000\].

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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 fundamental concept in algebra. It allows you to multiply a term outside the parentheses by each term inside the parentheses.
In our exercise, we see it applied to solve the equation: \(0.035x + 0.052(x + 5000) = 1130\).
We start by distributing \(0.052\) to both terms inside the parentheses:
\[0.052(x + 5000) = 0.052x + 0.052 \times 5000\]
This distributes the 0.052 to \(x\) and 5000 individually.
  • First, multiply \(0.052\) by \(x\) to get \(0.052x\).
  • Next, multiply \(0.052\) by 5000 to get 260.

So, the distribution simplifies to \(0.052x + 260\). Now the equation looks like: \(0.035x + 0.052x + 260 = 1130\).
Combining Like Terms
Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable.
In our example, we have the equation: \(0.035x + 0.052x + 260 = 1130\).
Here, \(0.035x\) and \(0.052x\) are like terms because they both have the variable \(x\).
We combine them by adding their coefficients:
  • Add \(0.035\) and \(0.052\) to get \(0.087\).
Now the equation is simplified to:
\(0.087x + 260 = 1130\).
This process helps reduce the complexity of the equation, making it easier to solve for the variable.
Isolating the Variable
Isolating the variable means getting the variable term alone on one side of the equation.
In our simplified equation \(0.087x + 260 = 1130\):
First, we subtract 260 from both sides to remove the constant term from the left side:
\[0.087x = 1130 - 260\]
This simplifies to:
\[0.087x = 870\].
The next step is to divide both sides by \(0.087\) to solve for \(x\):
\[x = \frac{870}{0.087}\].
Finally, calculate the division to find:
  • \(x = 10,000\).

So, \(x\) equals 10,000. Isolating the variable is crucial because it allows us to find the value of the variable that solves the equation.

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Most popular questions from this chapter

In Exercises \(25-60\) solve the problem by first setting up a proportion or an equation. Round off your answers to the nearest hundredth. A currency exchange in Tokyo converts \(\$ 15\) U.S. into 1625.1 Japanese yen. Find the yen value of \(\$ 200\) U.S.

In Exercises \(19-24,\) solve each proportion. Round off your answers to the nearest hundredth where necessary. $$\frac{x}{0.43}=\frac{0.26}{0.9}$$

Solve the equations and inequalities. $$0.6(x-4)-0.4(x-5)=4.6$$

Reduce each of the following rational expressions to lowest terms. $$\frac{\left(4 y^{4}\right)\left(5 y^{5}\right)}{\left(8 y^{3}\right)\left(y^{6}\right)}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{4 y}{3 x^{2}}-\frac{3}{2 x}+\frac{y}{x^{2}}$$
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