Solve the following equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

x − 14
x + 1
−
x
x − 1
= 0
x =

The equation is:

\[
\frac{x - 14}{x + 1} - \frac{x}{x - 1} = 0
\]

Let's solve this step by step.

### Step 1: Combine the fractions
The first step is to find a common denominator for the two fractions. The denominators are \(x + 1\) and \(x - 1\). The common denominator will be \((x + 1)(x - 1)\).

Thus, we rewrite the equation as:

\[
\frac{(x - 14)(x - 1)}{(x + 1)(x - 1)} - \frac{x(x + 1)}{(x + 1)(x - 1)} = 0
\]

### Step 2: Combine the numerators
Now that the denominators are the same, we can combine the numerators:

\[
\frac{(x - 14)(x - 1) - x(x + 1)}{(x + 1)(x - 1)} = 0
\]

### Step 3: Simplify the numerator
Expand both terms in the numerator:

1. \((x - 14)(x - 1) = x(x - 1) - 14(x - 1) = x^2 - x - 14x + 14 = x^2 - 15x + 14\)
2. \(x(x + 1) = x^2 + x\)

Substitute these into the equation:

\[
\frac{(x^2 - 15x + 14) - (x^2 + x)}{(x + 1)(x - 1)} = 0
\]

Simplify the numerator:

\[
x^2 - 15x + 14 - x^2 - x = -16x + 14
\]

Now, the equation becomes:

\[
\frac{-16x + 14}{(x + 1)(x - 1)} = 0
\]

### Step 4: Solve the equation
The fraction equals 0 when the numerator equals 0. So, solve the equation:

\[
-16x + 14 = 0
\]

\[
-16x = -14
\]

\[
x = \frac{14}{16} = \frac{7}{8}
\]

### Step 5: Check for restrictions
The original equation has restrictions that \(x \neq -1\) and \(x \neq 1\) because these values would make the denominators zero.

Since \(x = \frac{7}{8}\) is not one of these restricted values, it is a valid solution.

### Final Answer:
\[
x = \frac{7}{8}
\]

Let me know if you'd like a more detailed explanation or have any questions! Here are 5 related problems:

1. Solve \(\frac{x - 5}{x + 2} - \frac{x}{x - 3} = 0\).
2. Simplify \(\frac{a - b}{a + b} - \frac{b}{a - b}\).
3. Find the roots of the equation \(\frac{x}{x + 1} - \frac{2x}{x - 1} = 0\).
4. Solve the equation \(\frac{2x + 3}{x - 2} = \frac{4}{x + 1}\).
5. Solve \(\frac{x^2 - 9}{x + 3} = 0\).

**Tip:** Always check for restrictions when working with rational equations to avoid dividing by zero.

Solve the following equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) x − 14 / (x + 1) − x / (x − 1) = 0, x = ?

Learn how to solve the rational equation (x - 14)/(x + 1) - x/(x - 1) = 0 with a step-by-step solution. This problem covers key algebraic concepts such as finding common denominators, factoring, and solving linear equations. Suitable for students in Grades 9-12.

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