https://file.aiquickdraw.com/mathbot/file/1725537394203uwy298i2.jpg

The equation in the image is:

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

Let's solve this equation step-by-step:

### Step 1: Eliminate the denominator
Multiply both sides of the equation by \(x - 1\) to get rid of the fraction:

\[
4x = 2(x - 1)
\]

### Step 2: Distribute on the right side
Distribute the 2 on the right side of the equation:

\[
4x = 2x - 2
\]

### Step 3: Move all terms involving \(x\) to one side
Subtract \(2x\) from both sides:

\[
4x - 2x = -2
\]

This simplifies to:

\[
2x = -2
\]

### Step 4: Solve for \(x\)
Divide both sides by 2:

\[
x = -1
\]

### Step 5: Check for restrictions
Since the original equation has a denominator \(x - 1\), we need to make sure the solution doesn't make the denominator zero. The denominator would be zero if \(x = 1\), which is not our solution. Therefore, \(x = -1\) is valid.

### Final answer:
\[
x = -1
\]

Would you like more details or have any questions?

Here are five related questions for practice:
1. What would happen if the right side of the equation was 3 instead of 2?
2. Can you solve \(\frac{3x}{x-2} = 5\)?
3. What are the restrictions on values for \(x\) in rational equations like this?
4. How would the solution change if the equation was \(\frac{4x}{x-1} = 0\)?
5. Can you solve \(\frac{5x}{x-3} = 10\)?

**Tip:** Always check for restrictions in rational equations to ensure your solution doesn't make the denominator zero!

Learn how to solve the rational equation 4x/(x-1) = 2 with a step-by-step guide. This problem involves key algebraic concepts, including rational equations and restrictions on the domain. Suitable for students in Grades 8-10.

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