Question

a student tried to solve this rational equation by cross multiplying. they made a mistake. do two things: explain the mistake that was made. find the correct solutions. (a correct answer without correct reasoning scores you 0 points) $\frac{x}{2x - 3}=\frac{3x}{x + 11}$ $x(x + 11)=3x(2x - 3)$ $x^{2}+11 = 6x^{2}-9$ $0 = 5x^{2}-20$ $20 = 5x^{2}$ $4 = x^{2}$ $x=pm2$

Explanation:

Step1: Identify the mistake

The student made a distribution error. When expanding $x(x + 11)$, it should be $x^{2}+11x$, not $x^{2}+11$. When expanding $3x(2x - 3)$, it should be $6x^{2}-9x$, not $6x^{2}-9$.

Step2: Correct cross - multiplication

Starting from $x(x + 11)=3x(2x - 3)$, expand to get $x^{2}+11x = 6x^{2}-9x$.

Step3: Rearrange to form a quadratic equation

Move all terms to one side: $6x^{2}-9x-x^{2}-11x = 0$, which simplifies to $5x^{2}-20x=0$.

Step4: Factor the quadratic equation

Factor out $5x$: $5x(x - 4)=0$.

Step5: Solve for x

Set each factor equal to zero. If $5x=0$, then $x = 0$. If $x - 4=0$, then $x=4$.

Answer:

The mistake was incorrect distribution during expansion. The correct solutions are $x = 0$ and $x = 4$.