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Chapter 2: Problem 25

Solve the rational equation. Check your solutions. $$\frac{2 x}{x-1}-\frac{3}{x}=2$$

Short Answer

Expert verified
The solution to the equation is \( x = 3 \).

Step by step solution

01

Combine Like Terms

Combine like terms on one side of the equal equation. To do this, the equation can be rewritten as \( \frac{2x}{x-1} - \frac{3}{x} - 2 = 0 \).
02

Find a Common Denominator

The next step will be to find a common denominator between \(x-1\) and \(x\), which is \(x(x-1)\). Multiply the entire equation by the common denominator in order to eliminate fractions. The equation becomes: \(2x^2 - 3(x-1) - 2x(x-1) = 0 \).
03

Simplify the Equation

Simplify the equation by performing the necessary operations. The equation becomes: \(2x^2 - 3x + 3 - 2x^2 + 2x = 0\). Combine like terms to get: \( -x + 3 = 0 \).
04

Solve for the variable

Now, isolate \(x\) by subtracting subtract 3 from both sides to get: \( -x = -3 \). Then multiply by -1 to get: \( x = 3 \).
05

Check the Solution

Substitute \(x = 3\) back into the original equation to see if both sides become equal: \( \frac{2(3)}{3-1} - \frac{3}{3} = 2 \Rightarrow 3 - 1 = 2. \) Since the solution checks, \(x = 3\) is a valid solution.

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Key Concepts

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

Common Denominator
When solving rational equations, finding a common denominator is crucial. Rational equations contain fractions with different denominators. To simplify and solve these equations, you need a shared denominator.

In the given exercise, the rational equation has terms like \( \frac{2x}{x-1} \) and \( \frac{3}{x} \). Here, the common denominator is \( x(x-1) \).
  • Multiply each term by this common denominator to clear the fractions.
  • This transforms the expression into a polynomial equation, making it easier to solve.
Remembering to find and use a common denominator helps in managing and simplifying complex rational equations effectively.
Solving Rational Equations
After finding a common denominator, the next step is solving the rational equation. The aim is to simplify the transformed equation to isolate the variable.

For our equation, \( 2x^2 - 3(x-1) - 2x(x-1) = 0 \), you simplify by expanding and combining like terms. This gives:
  • \( 2x^2 - 3x + 3 - 2x^2 + 2x = 0 \)
  • Combine to get \( -x + 3 = 0 \)
Then solve for \( x \). Isolate \( x \) by moving terms around to find \( x = 3 \).
Solving these equations may vary in complexity, but systematically reducing and simplifying is key.
Verifying Solutions
Verifying solutions is an essential final step in solving rational equations. After finding a solution, substitute it back into the original equation to ensure correctness.

For \( x = 3 \), substitute back into \( \frac{2(3)}{3-1} - \frac{3}{3} \):
  • Calculate to verify both sides equal 2.
  • If they match, \( x = 3 \) is confirmed as correct.
Verification checks for errors. Sometimes, restrictions from denominators can invalidate potential solutions. Always validate to avoid such pitfalls.

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Most popular questions from this chapter

The point (2,4) on the graph of \(f(x)=x^{2}\) has been shifted horizontally to the point \((-3,4) .\) Identify the shift and write a new function \(g(x)\) in terms of \(f(x)\).

A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available. (a) Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) Write an expression for the area of the plot in terms of its length. (d) Find the dimensions that will yield the maximum area.

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In Exercises \(101-104,\) let \(f(t)=3 t+1\) and \(g(x)=x^{2}+4\). Find an expression for \((f \circ f)(t)\), and give the domain of \(f \circ f\)

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