Question

Solve the equation by the method of your choice.
StartFraction 1 Over x EndFraction plus StartFraction 1 Over x plus 4 EndFraction equals one half
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Part 1
The solution set is ​{
  
enter your response here​}.

Answer 1

To solve the equation

\[
\frac{1}{x} + \frac{1}{x + 4} = \frac{1}{2}
\]

we can start by finding a common denominator for the left side of the equation. The common denominator for \(x\) and \(x + 4\) is \(x(x + 4)\).

Rewriting each term with the common denominator gives:

\[
\frac{(x + 4) + x}{x(x + 4)} = \frac{1}{2}
\]

Simplifying the numerator:

\[
\frac{2x + 4}{x(x + 4)} = \frac{1}{2}
\]

Next, we cross-multiply to eliminate the fractions:

\[
2(2x + 4) = 1 \cdot x(x + 4)
\]

This simplifies to:

\[
4x + 8 = x^2 + 4x
\]

Now, we can subtract \(4x\) from both sides to simplify the equation further:

\[
8 = x^2
\]

This can be rearranged to:

\[
x^2 - 8 = 0
\]

Now we can factor this quadratic equation:

\[
(x - \sqrt{8})(x + \sqrt{8}) = 0
\]

The solutions are:

\[
x = \sqrt{8} \quad \text{or} \quad x = -\sqrt{8}
\]

Simplifying \(\sqrt{8}\) gives us:

\[
x = 2\sqrt{2} \quad \text{and} \quad x = -2\sqrt{2}
\]

The solution set is:

\[
\{2\sqrt{2}, -2\sqrt{2}\}
\]

Thus, the final answer is

\[
\{2\sqrt{2}, -2\sqrt{2}\}.
\]