Problem 25 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

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

Step 1: Combine Like Terms

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

Step 2: 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: \(2x^2 - 3(x-1) - 2x(x-1) = 0 \).

Step 3: 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: \( -x + 3 = 0 \).

Step 4: Solve for the variable

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

Step 5: 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.

Key Concepts

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.