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 becomes \((2x+5)(x-2)\), the second term becomes \(-6x\), and the right side becomes \$2x(x-2)$.
5. Expand all products and simplify the resulting equation to form a polynomial equation, then collect like terms and solve for \(x\) using appropriate algebraic methods (factoring, quadratic formula, etc.).

Key Concepts

Solving Rational Equations

Rational equations involve expressions with variables in the denominator. To solve them, first find a common denominator to combine terms or clear denominators by multiplying both sides. This helps eliminate fractions and simplifies the equation to a polynomial form.

Finding and Excluding Restrictions

When solving equations with variables in denominators, identify values that make any denominator zero, as these are restrictions and cannot be solutions. Excluding these values ensures the solution set is valid and avoids undefined expressions.

Solving Linear Equations

After clearing denominators, the resulting equation is often linear. Use algebraic techniques like combining like terms and isolating the variable to find the solution. Verify the solution by substituting back into the original equation.