X+1 over X-4=x-1 over x solving rational equation 

We are given the equation:

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

### Step 1: Cross-multiply
To eliminate the fractions, we will cross-multiply:

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

### Step 2: Expand both sides
Let's expand both sides of the equation:

\[
(x + 1) \cdot x = x^2 + x
\]
\[
(x - 1) \cdot (x - 4) = x^2 - 4x - x + 4 = x^2 - 5x + 4
\]

Now, the equation becomes:

\[
x^2 + x = x^2 - 5x + 4
\]

### Step 3: Simplify the equation
Subtract \(x^2\) from both sides:

\[
x = -5x + 4
\]

### Step 4: Solve for \(x\)
Now, solve for \(x\):

\[
x + 5x = 4
\]
\[
6x = 4
\]
\[
x = \frac{4}{6} = \frac{2}{3}
\]

### Step 5: Check for extraneous solutions
The original equation has \(x\) in the denominators, so we need to check if \(x = \frac{2}{3}\) makes any denominator zero. Since neither \(x - 4\) nor \(x\) becomes zero when \(x = \frac{2}{3}\), this is a valid solution.

Thus, the solution to the equation is:

\[
x = \frac{2}{3}
\]

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

Here are 5 related questions for practice:
1. Solve \(\frac{x+2}{x-3} = \frac{x-2}{x}\).
2. Solve \(\frac{x-1}{x+4} = \frac{2x}{x+5}\).
3. Solve \(\frac{x+3}{x-2} = \frac{5}{x}\).
4. Solve \(\frac{2x-1}{x+1} = \frac{3x}{x-2}\).
5. Solve \(\frac{x+4}{x-6} = \frac{x-3}{x+2}\).

**Tip**: Always check for extraneous solutions by ensuring the variable doesn't make any denominator zero.

X+1 over X-4 = x-1 over x solving rational equation

Learn how to solve the rational equation X+1 over X-4 = X-1 over X using cross-multiplication and linear equation techniques. This detailed step-by-step solution covers rational equations and checking for extraneous solutions, ideal for students in Grades 8-10.

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