Answer
$x = -\dfrac{1}{12}$
Work Step by Step
The least common denominator, or LCD, is $6x$.
Convert each fraction to an equivalent one by multiplying its numerator with whatever factor is missing between its denominator and the LCD:
$$\begin{align*}
\dfrac{3(3)}{6x} - \dfrac{5(2)}{6x} &= \dfrac{12x}{6x}\\
\\\dfrac{9}{6x} - \dfrac{10}{6x} &= \dfrac{12x}{6x}
\end{align*}$$
Subtract the fractions:
$$\frac{-1}{6x} = \frac{12x}{6x}$$
Multiply each side of the equation by $6x$ to eliminate the fractions:
$$-1 = 12x$$
Divide each side of the equation by $12$:
$$-\frac{1}{12}=x$$
To check the solution, plug in the values we just found for $x$ into the original equation:
$$\dfrac{3}{2(-\frac{1}{12})} - \dfrac{5}{3(-\frac{1}{12})} = 2$$
Simplify the fractions:
$$\frac{3}{-\frac{2}{12}} - \frac{5}{-\frac{3}{12}} = 2$$
Use the rule $\dfrac{a}{\frac{b}{c}}=a \cdot \frac{c}{b}$ to obtain:
$$\begin{align*}
3\cdot\frac{12}{-2}-5\cdot\frac{12}{-3}&=2\\
-\frac{36}{2} + \frac{60}{3} &= 2\\
-18+20&=2\\
2&=2\end{align*}$$
Both sides are equal to one another; therefore, this solution is correct.
