We are tasked with solving the equation:

\[
\frac{20}{x} - \frac{20}{x-2} = \frac{4}{x}
\]

### Step 1: Eliminate the fractions
We want to get rid of the denominators by multiplying both sides of the equation by the least common denominator (LCD). The LCD of \(x\) and \(x - 2\) is \(x(x-2)\).

Multiply both sides of the equation by \(x(x-2)\):

\[
x(x-2) \left( \frac{20}{x} - \frac{20}{x-2} \right) = x(x-2) \cdot \frac{4}{x}
\]

### Step 2: Simplify
Now, distribute the \(x(x-2)\) to each term:

\[
x(x-2) \cdot \frac{20}{x} - x(x-2) \cdot \frac{20}{x-2} = x(x-2) \cdot \frac{4}{x}
\]

Simplifying each term:

- First term: \(\frac{20}{x} \cdot x(x-2) = 20(x-2)\)
- Second term: \(\frac{20}{x-2} \cdot x(x-2) = 20x\)
- Right-hand side: \(\frac{4}{x} \cdot x(x-2) = 4(x-2)\)

Now the equation becomes:

\[
20(x-2) - 20x = 4(x-2)
\]

### Step 3: Expand and simplify
Expand both sides:

\[
20x - 40 - 20x = 4x - 8
\]

Simplify the left-hand side:

\[
-40 = 4x - 8
\]

### Step 4: Solve for \(x\)
Add 8 to both sides:

\[
-40 + 8 = 4x
\]

\[
-32 = 4x
\]

Now, divide both sides by 4:

\[
x = -8
\]

### Final answer:
\[
x = -8
\]

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

---

Here are some related questions for further practice:

1. Solve: \(\frac{15}{x} - \frac{10}{x-3} = \frac{5}{x}\)
2. Solve: \(\frac{12}{x+1} - \frac{8}{x-2} = \frac{3}{x+1}\)
3. Solve: \(\frac{25}{x} - \frac{10}{x-5} = \frac{15}{x+5}\)
4. Solve: \(\frac{18}{x} + \frac{6}{x+3} = \frac{12}{x-4}\)
5. Solve: \(\frac{10}{x} - \frac{5}{x-1} = \frac{5}{x-3}\)

**Tip:** Always try to factor and simplify the equation before multiplying or dividing terms, as this can save time and reduce errors.

Learn how to solve the rational equation 20/x - 20/(x-2) = 4/x step-by-step. This detailed guide covers key algebraic concepts such as solving rational equations, eliminating fractions, and simplifying expressions. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation