Find the solution set (2)/(x-1)+4-(14)/(x-1)

We are tasked with solving the equation:

\[
\frac{2}{x - 1} + 4 - \frac{14}{x - 1} = 0
\]

### Step 1: Combine like terms

Both \(\frac{2}{x-1}\) and \(\frac{14}{x-1}\) have the same denominator, so we can combine them.

\[
\frac{2 - 14}{x - 1} + 4 = 0
\]

This simplifies to:

\[
\frac{-12}{x - 1} + 4 = 0
\]

### Step 2: Isolate the fraction

Subtract 4 from both sides to isolate the fraction:

\[
\frac{-12}{x - 1} = -4
\]

### Step 3: Solve for \(x\)

Multiply both sides by \((x - 1)\) to eliminate the denominator:

\[
-12 = -4(x - 1)
\]

Simplify:

\[
-12 = -4x + 4
\]

### Step 4: Rearrange the equation

Move all terms involving \(x\) to one side:

\[
-4x = -12 - 4
\]

\[
-4x = -16
\]

### Step 5: Solve for \(x\)

Divide both sides by \(-4\):

\[
x = 4
\]

### Solution set:
The solution to the equation is \(x = 4\).

Let me know if you'd like more details or have any questions!

Here are 5 related questions to expand on this concept:

1. How do you solve similar rational equations with different denominators?
2. What happens when a solution leads to an undefined expression, like division by zero?
3. How can you check if a solution satisfies the original equation?
4. What are the steps to combine fractions with unlike denominators?
5. How do extraneous solutions arise in rational equations?

**Tip:** Always check your solution by substituting it back into the original equation to ensure it's valid!

Find the solution set (2)/(x-1) + 4 - (14)/(x-1)

This step-by-step guide explains how to solve the rational equation 2/(x-1) + 4 - 14/(x-1) = 0. Learn to combine fractions and isolate variables in linear equations, suitable for students in Grades 8-10.

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