$$ \frac{1}{x}-\frac{1}{x+1}=\frac16 $$solve for x

We are given the equation:

\[
\frac{1}{x} - \frac{1}{x+1} = \frac{1}{6}
\]

### Step 1: Combine the terms on the left-hand side by finding a common denominator
The common denominator for $ x $ and $ x+1 $ is $ x(x+1) $. Rewriting the terms:

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

Now the equation becomes:

\[
\frac{1}{x(x+1)} = \frac{1}{6}
\]

### Step 2: Cross multiply
To eliminate the fractions, we cross multiply:

\[
1 \cdot 6 = 1 \cdot x(x+1)
\]

\[
6 = x(x+1)
\]

### Step 3: Expand and simplify the quadratic equation
Expand the right-hand side:

\[
6 = x^2 + x
\]

Rearranging the equation:

\[
x^2 + x - 6 = 0
\]

### Step 4: Solve the quadratic equation
We now solve $ x^2 + x - 6 = 0 $ using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation $ x^2 + x - 6 = 0 $, the coefficients are:
- $ a = 1 $
- $ b = 1 $
- $ c = -6 $

Substituting into the quadratic formula:

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-6)}}{2(1)}
\]

\[
x = \frac{-1 \pm \sqrt{1 + 24}}{2}
\]

\[
x = \frac{-1 \pm \sqrt{25}}{2}
\]

\[
x = \frac{-1 \pm 5}{2}
\]

### Step 5: Calculate the solutions
Now, we compute the two possible solutions:

1. $ x = \frac{-1 + 5}{2} = \frac{4}{2} = 2 $
2. $ x = \frac{-1 - 5}{2} = \frac{-6}{2} = -3 $

### Step 6: Verify the solutions
We need to check that neither solution makes the original equation undefined. The only issue would be if $ x = -1 $, which would make one of the denominators zero. Neither solution does this, so both solutions are valid.

Thus, the solutions are:

\[
x = 2 \quad \text{or} \quad x = -3
\]

Would you like more details on any step or have any questions?

Here are five related questions for practice:
1. Solve $ \frac{1}{x} - \frac{1}{x+2} = \frac{1}{4} $.
2. Solve the equation $ \frac{2}{x} - \frac{1}{x+1} = \frac{1}{3} $.
3. Factor the quadratic $ x^2 + x - 6 = 0 $ directly without using the quadratic formula.
4. Solve the equation $ \frac{1}{x} + \frac{1}{x+1} = \frac{1}{2} $.
5. Solve $ \frac{2}{x} - \frac{3}{x+2} = \frac{5}{12} $.

**Tip**: Always check the domain of the solution to avoid dividing by zero, which makes the function undefined.

$$ \frac{1}{x}-\frac{1}{x+1} = \frac{1}{6} $$ solve for x

Learn how to solve the rational equation $$ \frac{1}{x} - \frac{1}{x+1} = \frac{1}{6} $$ by finding a common denominator and using the quadratic formula. This step-by-step solution covers key algebraic concepts like solving quadratic equations. Suitable for students in Grades 9-11.

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