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 sixth
Question content area bottom
Part 1
The solution set is {
enter your response here}.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)
StartFraction 1 Over x EndFraction plus StartFraction 1 Over x plus 4 EndFraction equals one sixth
Question content area bottom
Part 1
The solution set is {
enter your response here}.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)
Answers
GPT-4o mini
To solve the equation
\[
\frac{1}{x} + \frac{1}{x + 4} = \frac{1}{6},
\]
we 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 the left side gives us:
\[
\frac{x + 4 + x}{x(x + 4)} = \frac{2x + 4}{x(x + 4)}.
\]
Thus, the equation becomes:
\[
\frac{2x + 4}{x(x + 4)} = \frac{1}{6}.
\]
Next, we cross-multiply:
\[
6(2x + 4) = x(x + 4).
\]
Expanding both sides, we have:
\[
12x + 24 = x^2 + 4x.
\]
Rearranging the equation to set it to zero, we get:
\[
x^2 + 4x - 12x - 24 = 0,
\]
which simplifies to:
\[
x^2 - 8x - 24 = 0.
\]
Now we can use the quadratic formula to solve for \(x\):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \(a = 1\), \(b = -8\), and \(c = -24\). Plugging in these values gives:
\[
x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1},
\]
\[
x = \frac{8 \pm \sqrt{64 + 96}}{2},
\]
\[
x = \frac{8 \pm \sqrt{160}}{2}.
\]
Since \(\sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}\), we can simplify further:
\[
x = \frac{8 \pm 4\sqrt{10}}{2},
\]
\[
x = 4 \pm 2\sqrt{10}.
\]
Thus, the solutions are:
\[
x = 4 + 2\sqrt{10} \quad \text{and} \quad x = 4 - 2\sqrt{10}.
\]
We can now express the solution set:
\[
\{ 4 + 2\sqrt{10}, 4 - 2\sqrt{10} \}.
\]
So, the final answer is:
\[
\{ 4 + 2\sqrt{10}, 4 - 2\sqrt{10} \}.
\]
\[
\frac{1}{x} + \frac{1}{x + 4} = \frac{1}{6},
\]
we 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 the left side gives us:
\[
\frac{x + 4 + x}{x(x + 4)} = \frac{2x + 4}{x(x + 4)}.
\]
Thus, the equation becomes:
\[
\frac{2x + 4}{x(x + 4)} = \frac{1}{6}.
\]
Next, we cross-multiply:
\[
6(2x + 4) = x(x + 4).
\]
Expanding both sides, we have:
\[
12x + 24 = x^2 + 4x.
\]
Rearranging the equation to set it to zero, we get:
\[
x^2 + 4x - 12x - 24 = 0,
\]
which simplifies to:
\[
x^2 - 8x - 24 = 0.
\]
Now we can use the quadratic formula to solve for \(x\):
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]
where \(a = 1\), \(b = -8\), and \(c = -24\). Plugging in these values gives:
\[
x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1},
\]
\[
x = \frac{8 \pm \sqrt{64 + 96}}{2},
\]
\[
x = \frac{8 \pm \sqrt{160}}{2}.
\]
Since \(\sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10}\), we can simplify further:
\[
x = \frac{8 \pm 4\sqrt{10}}{2},
\]
\[
x = 4 \pm 2\sqrt{10}.
\]
Thus, the solutions are:
\[
x = 4 + 2\sqrt{10} \quad \text{and} \quad x = 4 - 2\sqrt{10}.
\]
We can now express the solution set:
\[
\{ 4 + 2\sqrt{10}, 4 - 2\sqrt{10} \}.
\]
So, the final answer is:
\[
\{ 4 + 2\sqrt{10}, 4 - 2\sqrt{10} \}.
\]