Solve each equation. See Example 1. (2x+5)/2 - 3x/(x-2) = x

Verified step by step guidance

1. Identify the equation to solve: \(\frac{2x+5}{2} - \frac{3x}{x-2} = x\).
2. Find the least common denominator (LCD) of the fractions, which is \$2(x-2)$, to eliminate the denominators by multiplying every term by this LCD.
3. Multiply each term by the LCD \$2(x-2)$ to clear the fractions: \(2(x-2) \times \frac{2x+5}{2} - 2(x-2) \times \frac{3x}{x-2} = 2(x-2) \times x\).
4. Simplify each term after multiplication: the first term simplifies to \((2x+5)(x-2)\), the second term simplifies to \(-6x\), and the right side becomes \$2x(x-2)$.
5. Expand all expressions and rearrange the equation to one side to form a polynomial equation, then solve for \(x\) using factoring or the quadratic formula.

Key Concepts

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, identify the least common denominator (LCD) to eliminate fractions by multiplying both sides, simplifying the equation to a polynomial form. Always check for restrictions where denominators become zero.

Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that all denominators divide into without remainder. It is used to clear fractions by multiplying each term, allowing the equation to be rewritten without denominators. For example, for denominators 2 and (x-2), the LCD is 2(x-2).

Checking for Extraneous Solutions

After solving the equation, substitute solutions back into the original denominators 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.