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

Verified step by step guidance

1. Start by finding a common denominator for the fractions on the left side of the equation. The denominators are 4 and (x+1), so the common denominator is 4(x+1).
2. Rewrite each term with the common denominator: $\frac{(4x+3)(x+1)}{4(x+1)}-\frac{2x\cdot 4}{4(x+1)}=x$.
3. Combine the fractions on the left side: $\frac{(4x+3)(x+1)-8x}{4(x+1)}=x$.
4. Clear the fraction by multiplying both sides of the equation by 4(x+1): $(4x+3)(x+1)-8x=x\cdot 4(x+1)$.
5. Expand and simplify both sides of the equation, then solve for x.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables. This includes operations such as adding, subtracting, multiplying, and dividing both sides of an equation, as well as combining like terms. Mastery of these techniques is essential for solving equations effectively.

Common Denominator

A common denominator is a shared multiple of the denominators in a set of fractions, allowing for the addition or subtraction of those fractions. In the context of the given equation, finding a common denominator is crucial for simplifying the expression and eliminating fractions, which makes the equation easier to solve.

Isolating the Variable

Isolating the variable means rearranging the equation so that the variable of interest is on one side and all other terms are on the opposite side. This process often involves inverse operations and is a fundamental step in solving equations, as it allows for direct calculation of the variable's value.