Answer
$x = 2 \text{ or } x = -1$
Work Step by Step
The least common denominator, or LCD, is $2x$, 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:
$$\dfrac{1(2)}{2x} + \dfrac{x(x)}{2x} = \dfrac{x + 4}{2x}$$
Multiply to simplify:
$$\dfrac{2}{2x} + \dfrac{x^2}{2x} = \dfrac{x + 4}{2x}$$
Add the fractions:
$$\dfrac{x^2 + 2}{2x} = \dfrac{x + 4}{2x}$$
Multiply each side of the equation by $2x$ to eliminate the fractions:
$$x^2 + 2 = x + 4$$
Move all terms to the left side of the equation:
$$\begin{align}
x^2+2-(x+4)&=0\\
x^2+2-x-4&=0\\
x^2 - x - 2 &= 0
\end{align}$$
Factor the expression:
$$(x - 2)(x + 1) = 0$$
Use the Zero-Product Property by setting each factor equal to $0$, then solve each equation for $x$:
First factor:
$x - 2 = 0$
Add $2$ to each side of the equation:
$x = 2$
Second factor:
$x + 1 = 0$
Subtract $1$ from each side of the equation:
$x = -1$
To check the solution, plug in the values we just found for $x$ into the original equation:
$\dfrac{1}{2} + \dfrac{2}{2} = \dfrac{2 + 4}{2(2)}$
Simplify the fractions:
$\dfrac{1}{2} + 1 = \dfrac{6}{4}$
Convert to equivalent fractions with $4$ as the LCD:
$\dfrac{2}{4} + \dfrac{4}{4} = \dfrac{6}{4}$
Add to simplify:
$\dfrac{6}{4} = \dfrac{6}{4}$
Both sides are equal to one another; therefore, this solution is correct.
Check the second solution:
$\dfrac{1}{-1} + \dfrac{(-1)}{2} = \dfrac{-1 + 4}{2(-1)}$
Simplify the fractions:
$-1 - \dfrac{1}{2} = \dfrac{3}{-2}$
Convert to equivalent fractions:
$-\dfrac{2}{2} - \dfrac{1}{2} = -\dfrac{3}{2}$
Add to simplify:
$-\dfrac{3}{2} = -\dfrac{3}{2}$
Both sides are equal to one another; therefore, this solution is correct.
