Solve each equation. (4x+3)/4 - 2x/(x+1) = x

Verified step by step guidance

1. Start by writing down the given equation: \(\frac{4x+3}{4} - \frac{2x}{x+1} = x\).
2. Identify the denominators in the equation, which are 4 and \(x+1\). To eliminate the fractions, find the least common denominator (LCD), which is \$4(x+1)$.
3. Multiply every term in the equation by the LCD \$4(x+1)$ to clear the denominators: \(4(x+1) \times \frac{4x+3}{4} - 4(x+1) \times \frac{2x}{x+1} = 4(x+1) \times x\).
4. Simplify each term after multiplication: the first term becomes \((4x+3)(x+1)\), the second term becomes \(4 \times 2x = 8x\), and the right side becomes \$4x(x+1)$.
5. Expand all products and simplify the resulting equation to form a polynomial equation. Then, collect like terms on one side to set the equation equal to zero, preparing it for solving by factoring or using the quadratic formula.

Key Concepts

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, find a common denominator to combine terms or clear denominators by multiplying both sides, ensuring to consider restrictions where denominators are zero.

Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that all denominators divide into evenly. Identifying the LCD allows you to eliminate fractions by multiplying through, simplifying the equation to a polynomial form that is easier to solve.

Checking for Extraneous Solutions

After solving, substitute solutions back into the original equation to ensure they do not make any denominator zero. Solutions that cause division by zero are extraneous and must be excluded from the final answer.