solve for x 1. 1/x=2/(3x-1)  2. 1/x + 1/(2x)=2/3

### 1. Solve \( \frac{1}{x} = \frac{2}{3x - 1} \)

To solve for \( x \):

Step 1: Cross-multiply to eliminate the fractions:
\[
1 \cdot (3x - 1) = 2 \cdot x
\]
This simplifies to:
\[
3x - 1 = 2x
\]

Step 2: Isolate \( x \) by subtracting \( 2x \) from both sides:
\[
3x - 2x = 1
\]
\[
x = 1
\]

Thus, the solution to the first equation is \( x = 1 \).

---

### 2. Solve \( \frac{1}{x} + \frac{1}{2x} = \frac{2}{3} \)

Step 1: Find a common denominator on the left-hand side:
\[
\frac{1}{x} + \frac{1}{2x} = \frac{2}{2x} + \frac{1}{2x} = \frac{3}{2x}
\]
Thus, the equation becomes:
\[
\frac{3}{2x} = \frac{2}{3}
\]

Step 2: Cross-multiply to eliminate the fractions:
\[
3 \cdot 3 = 2x \cdot 2
\]
\[
9 = 4x
\]

Step 3: Solve for \( x \):
\[
x = \frac{9}{4}
\]

Thus, the solution to the second equation is \( x = \frac{9}{4} \).

---

### Solutions:
1. \( x = 1 \)
2. \( x = \frac{9}{4} \)

Would you like more details or have any questions? 

Here are 5 related questions you might explore:
1. What happens if \( x = 0 \) in equations involving fractions?
2. How can cross-multiplication help in solving rational equations?
3. How do you handle more complex rational expressions in equations?
4. What methods are used to check for extraneous solutions in rational equations?
5. What is the importance of finding a common denominator when adding fractions?

**Tip:** Always check your solutions by plugging them back into the original equation to ensure they satisfy the equation!

solve for x 1. 1/x=2/(3x-1) 2. 1/x + 1/(2x)=2/3

Explore how to solve rational equations like 1/x = 2/(3x-1) and 1/x + 1/(2x) = 2/3 using cross multiplication and common denominators. This detailed solution is perfect for students in Grades 9-10 learning key algebraic concepts.

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