Problem 8 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\frac{4 x+3}{4}-\frac{2 x}{x+1}=x$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{5}\).

Step 1: Eliminate the fractions

To eliminate the fractions in the equation \(\frac{4x+3}{4} - \frac{2x}{x+1} = x\), find a common denominator. The common denominator for 4 and \(x+1\) is \(4(x+1)\). Multiply the whole equation by \(4(x+1)\).

Step 2: Distribute and simplify

Multiply each term by \(4(x+1)\): \[4(x+1) \frac{4x+3}{4} - 4(x+1) \frac{2x}{x+1} = 4(x+1) x\]. Simplify each multiplication: \[ (4x+3)(x+1) - 8x = 4x(x+1)\].

Step 3: Expand the equations

Expand each term: \[4x(x+1) + 3(x+1) - 8x = 4x^2 + 4x\] becomes \[4x^2 + 4x + 3x + 3 - 8x = 4x^2 + 4x\].

Step 4: Combine like terms

Combine like terms on the left side: \[4x^2 - 1x + 3 = 4x^2 + 4x\].

Step 5: Move all terms to one side

Move all terms to one side to set the equation to zero: \[4x^2 - 4x^2 - x + 3 - 4x = 0\] simplifies to \[-5x + 3 = 0\].

Step 6: Solve for x

Isolate \(x\) by moving constant terms to the opposite side and divide: \[-5x = -3 \) then \(x = \frac{3}{5}\].

Key Concepts

Eliminating Fractions

When solving rational equations, fractions can make the process more complex. To simplify, eliminate them. Start by identifying a common denominator for all the fractions. In the given exercise, the denominators are 4 and \(x+1\). The common denominator here would be \(4(x+1)\). Multiply every term in the equation by this common denominator to clear the fractions. This step is crucial for making the equation simpler and easier to handle.

Common Denominator

Finding a common denominator allows you to combine and eliminate fractions. For our equation, the common denominator between 4 and \(x+1\) is \(4(x+1)\). By multiplying each term by \(4(x+1)\), you standardize the denominators, enabling the fractions to cancel out. This transformation makes the equation: \[4(x+1) \cdot \frac{4x+3}{4} - 4(x+1) \cdot \frac{2x}{x+1} = 4(x+1) \cdot x\].

Combining Like Terms

Combining like terms is a fundamental algebraic process to simplify equations. After eliminating the fractions and expanding the equation, group similar terms together. In our provided example: \(4x^2 + 4x + 3x + 3 - 8x = 4x^2 + 4x\), combine the \(x\)-terms on the left side: \(4x^2 - 1x + 3\). This simplifies the equation because fewer terms allow for easier manipulation and solving.

Expanding Equations

Expanding equations involves using distributive properties to remove parentheses, making each term explicit. This step is performed after eliminating the fractions and before combining like terms. Take each term and multiply them accordingly: \((4x+3)(x+1) - 8x = 4x(x+1)\). Fully expand to \(4x^2 + 4x + 3x + 3 - 8x\). Expansion transforms the equation into a polynomial form, which is easier to solve via traditional algebraic methods.