Let's solve the equation:

\[
\frac{6}{x} + \left( \frac{3}{x} + 2 \right) = 2
\]

### Step 1: Combine like terms
First, simplify the left side by combining the two fractions with \( \frac{6}{x} \) and \( \frac{3}{x} \):

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

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

\[
\frac{9}{x} + 2 = 2
\]

### Step 2: Isolate the term with \( x \)
Now, subtract 2 from both sides:

\[
\frac{9}{x} = 2 - 2
\]

\[
\frac{9}{x} = 0
\]

### Step 3: Solve for \( x \)
For \( \frac{9}{x} = 0 \), there is no solution since \( 9 \neq 0 \), and thus \( x \) cannot make this expression true.

### Conclusion:
The equation has **no solution**.

Would you like any more details or have any questions? Here are 5 related questions for practice:
1. How would you solve \( \frac{4}{x} + 3 = 5 \)?
2. What happens when an equation has no solution?
3. Solve \( \frac{2}{x} + \frac{5}{x} = 3 \).
4. What if the equation was \( \frac{6}{x} + 3 = 2 \)?
5. How can you check if an equation has infinite solutions?

**Tip:** When you get an equation like \( \frac{a}{x} = 0 \), remember that there's no solution because the numerator must be 0, which only happens when \( a = 0 \).

Explore how to solve the rational equation 6/x + (3/x + 2) = 2 with a detailed explanation. This problem illustrates key algebraic concepts such as combining like terms and understanding when an equation has no solution. Suitable 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