Question 261·Hard·Math

Dividing by $0$ is undefined, so the shared denominator cannot be $0$. Find the input it rules out:

Add $5x$ to both sides:

Divide by $5$:

So $x=\frac35$ cannot be a solution.

Type the original equation with every $x$ changed to $x_1$ and $=$ changed to $\sim$:

Desmos uses $\sim$ to fit $x_1$ instead of graphing $x$. Under PARAMETERS, it shows $x_1\approx0.619047619$. That's a candidate, not yet an exact answer.

Type $x_1$ on a new line and press its fraction button. Desmos shows $\frac{13}{21}$. This is about $0.619$, while the excluded $\frac35$ is $0.6$, so the candidate is allowed. The value of $x$ is $\frac{13}{21}$. Choice D.

For the allowed inputs $x\ne\frac35$, multiply every term on both sides by $3-5x$:

The $+2$ and $-1$ get multiplied too. Leaving either constant alone changes the equation.

Multiply each term inside the parentheses, keeping the minus sign on the right:

Combine the number terms and the $x$-terms on each side:

In particular, $2(-5x)=-10x$, not $-8x$.

Add $10x$ to both sides:

This keeps all the $x$-terms together.

Add $3$ to both sides:

Now only multiplication separates $x$ from its value.

Divide by $21$:

This isn't the forbidden $\frac35$, so it solves the original equation. The value of $x$ is $\frac{13}{21}$. Choice D.