Questions: Solve the following rational equation and simplify your answer. -3x/(x+2) + 2/(2x-3) = -3

Transcript text: Solve the following rational equation and simplify your answer. \[ \frac{-3 x}{x+2}+\frac{2}{2 x-3}=-3 \]

Solution

Solution Steps

To solve the given rational equation, we first find a common denominator for the fractions on the left side of the equation. Then, we combine the fractions into a single fraction. Next, we set the equation equal to zero by moving all terms to one side. Finally, we solve for \( x \) by finding the roots of the resulting equation.

Step 1: Rewrite the Equation

We start with the rational equation: \[ \frac{-3x}{x+2} + \frac{2}{2x-3} = -3 \]

Step 2: Combine the Fractions

To combine the fractions, we find a common denominator, which is \((x + 2)(2x - 3)\). Rewriting the equation gives: \[ \frac{-3x(2x - 3) + 2(x + 2)}{(x + 2)(2x - 3)} = -3 \]

Step 3: Clear the Denominator

Multiplying both sides by \((x + 2)(2x - 3)\) results in: \[ -3x(2x - 3) + 2(x + 2) = -3(x + 2)(2x - 3) \]

Step 4: Expand and Simplify

Expanding both sides leads to: \[ -6x^2 + 9x + 2x + 4 = -3(2x^2 - 3x + 4x - 6) \] This simplifies to: \[ -6x^2 + 11x + 4 = -6x^2 + 9x + 18 \]

Step 5: Set the Equation to Zero

Rearranging gives: \[ 11x + 4 - 9x - 18 = 0 \] which simplifies to: \[ 2x - 14 = 0 \]

Step 6: Solve for \( x \)

Solving for \( x \) yields: \[ 2x = 14 \implies x = 7 \]