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Chapter 0: Problem 23

Solve the equation and check your solution. $$ \frac{3 x}{2}+\frac{1}{4}(x-2)=10 $$

Short Answer

Expert verified
The solution to the equation is \(x = 6\).

Step by step solution

01

Distribute the fraction

Begin by distributing the \(\frac{1}{4}\) to the terms inside the parentheses \((x-2)\). This will yield the equation: \(\frac{3x}{2} + \frac{1}{4}x - \frac{1}{2} = 10\)
02

Combine like terms

Add the \(x\) terms together and subtract \(\frac{1}{2}\) from both sides of the equation to isolate the \(x\) terms. \(\frac{3x + \frac{1}{4}x}{2} = 10 + \frac{1}{2}\) simplifies to \(\frac{7x}{4} = 10\frac{1}{2}\)\}
03

Isolate the variable \(x\)

Multiply both sides of the equation by \(\frac{4}{7}\) to isolate \(x\). \(\frac{4}{7} *\frac{7x}{4}\) = \(\frac{4}{7}*10\frac{1}{2}\) gives \(x = 6\)
04

Check the solution

Substitute the value of \(x\) back into the original equation to verify the solution. \(\frac{3(6)}{2} + \frac{1}{4}(6-2)\) equals 10 as required.

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Key Concepts

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

Understanding the Distributive Property
The distributive property is an essential tool in algebra that simplifies expressions. In this context, we distribute a fraction across terms inside parentheses. For the given equation \( \frac{3x}{2} + \frac{1}{4}(x-2) = 10 \), we must apply the distributive property to \( \frac{1}{4}(x-2) \). This involves multiplying \( \frac{1}{4} \) by each term inside the parentheses, resulting in \( \frac{1}{4}x - \frac{1}{2} \).

By distributing, we remove parentheses and make the equation easier to handle. This allows us to focus on simplifying and solving the rest of the equation.
Combining Like Terms
Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable component. After distributing in the equation, we have \( \frac{3x}{2} + \frac{1}{4}x - \frac{1}{2} = 10 \).

Here, \( \frac{3x}{2} \) and \( \frac{1}{4}x \) are like terms because they both contain the variable \( x \). To combine these, you might need a common denominator.
  • The term \( \frac{3x}{2} \) can be rewritten with a denominator of 4 as \( \frac{6x}{4} \).
  • Now combine: \( \frac{6x}{4} + \frac{1x}{4} = \frac{7x}{4} \).


This process simplifies the equation further and is a crucial step towards solving for the variable.
Isolating the Variable
To find the value of \( x \), isolating the variable is key. After simplifying via combining like terms, you get \( \frac{7x}{4} = 10 \frac{1}{2} \) in our equation. Isolating \( x \) involves transforming this into an equivalent expression where \( x \) stands alone on one side of the equation.

Multiply both sides by the reciprocal of \( \frac{7}{4} \), which is \( \frac{4}{7} \), to achieve this.
  • Multiplying gives \( x = \left(\frac{4}{7}\right) \left(10 \frac{1}{2}\right) \).
  • Convert \( 10 \frac{1}{2} \) to an improper fraction, \( \frac{21}{2} \), and calculate: \( x = \frac{4}{7} * \frac{21}{2} \), resulting in \( x = 6 \).


This method ensures we accurately determine the value of \( x \) in the equation.
Checking Your Solutions
After solving for \( x \), it's crucial to check your work to ensure accuracy. Substitute \( x = 6 \) back into the original equation \( \frac{3x}{2} + \frac{1}{4}(x-2) = 10 \). This verification step confirms whether our solution satisfies the equation.

Calculate each part:
  • Substitute to get \( \frac{3(6)}{2} + \frac{1}{4}(6-2) \).
  • Simplify: \( \frac{18}{2} + \frac{1}{4}(4) = 9 + 1 \).
  • The result is 10, matching the original equation's right side.


Checking solutions like this one provides assurance that our steps and results are correct, completing our problem-solving process.

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Most popular questions from this chapter

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ g(x)=\frac{x}{8} $$

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(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 120 feet. $$ t_{1}=0, t_{2}=2 $$

The total numbers \(f\) (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time (in years) is given by \(t\), with \(t=5\) corresponding to 1995 . (Source: U.S. Federal Highway Administration) $$ \begin{array}{|c|c|} \hline 0 \text { Year, } t & \text { Miles traveled, } f(t) \\ \hline 5 & 2423 \\ 6 & 2486 \\ 7 & 2562 \\ 8 & 2632 \\ 9 & 2691 \\ 10 & 2747 \\ 11 & 2797 \\ 12 & 2856 \\ \hline \end{array} $$ (a) Does \(f^{-1}\) exist? (b) If \(f^{-1}\) exists, what does it mean in the context of the problem? (c) If \(f^{-1}\) exists, find \(f^{-1}\) (2632). (d) If the table was extended to 2003 and if the total number of miles traveled by motor vehicles for that year was 2747 billion, would \(f^{-1}\) exist? Explain.
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