Answer
$x = -3 \text{ or }x = -2$
Work Step by Step
The least common denominator, or LCD, is $x$, in this case. Convert each fraction to an equivalent one by multiplying its numerator with whatever factor is missing between its denominator and the LCD:
$$\begin{align*}
\dfrac{x(x)}{x} + \dfrac{6}{x} &= -\dfrac{5(x)}{x}\\
\\\dfrac{x^2}{x} + \dfrac{6}{x} &= -\dfrac{5x}{x}\\
\\\dfrac{x^2 + 6}{x} &= -\dfrac{5x}{x}
\end{align*}$$
Multiply each side of the equation by $x$ to eliminate the fractions:
$$x^2 + 6 = -5x$$
Move all terms to the left side of the equation:
$$x^2 + 5x + 6 = 0$$
Factor the trinomial on the left side of the equation by looking for factors of $6$ whose sum is $5$:
$$(x + 3)(x + 2) = 0$$
Use the Zero-Product Property by equating each factor to zero, then solve each equation for $x$:
First factor:
$$\begin{align*}
x + 3 &= 0\\
x&=-3
\end{align*}$$
Second factor:
$$\begin{align*}
x + 2 &= 0\\
x&=-2
\end{align*}$$
To check the solution, plug in the values we just found for $x$ into the original equation:
First solution:
$-3 + \frac{6}{-3} = -5$
Simplify the fractions:
$-3 - 2 = -5$
Simplify:
$-5 = -5$
Both sides are equal to one another; therefore, this solution is correct.
Second solution:
$-2 + \frac{6}{-2} = -5$
Simplify the fractions:
$-2 - 3 = -5$
Simplify:
$-5 = -5$
Both sides are equal to one another; therefore, this solution is correct.
