Solve: 1/6x = 2x+5/3x² - 1/6x² If final answer is a fraction, write as a simplified fraction, #/#. NO DECIMALS. X =

The given equation is:

\[
\frac{1}{6x} = \frac{2x + 5}{3x^2} - \frac{1}{6x^2}
\]

### Step 1: Eliminate the denominators
To simplify, multiply both sides of the equation by the least common denominator (LCD). The LCD of \(6x\), \(3x^2\), and \(6x^2\) is \(6x^2\).

Multiply each term by \(6x^2\):

\[
6x^2 \cdot \frac{1}{6x} = 6x^2 \cdot \left( \frac{2x + 5}{3x^2} - \frac{1}{6x^2} \right)
\]

### Step 2: Simplify each term
- On the left-hand side:

\[
6x^2 \cdot \frac{1}{6x} = x
\]

- On the right-hand side:
  - First term: 
  \[
  6x^2 \cdot \frac{2x + 5}{3x^2} = 2(2x + 5) = 4x + 10
  \]
  - Second term: 
  \[
  6x^2 \cdot \frac{1}{6x^2} = 1
  \]

Now, the equation becomes:

\[
x = (4x + 10) - 1
\]

### Step 3: Combine like terms
Simplify the right-hand side:

\[
x = 4x + 9
\]

### Step 4: Isolate \(x\)
Move \(4x\) to the left-hand side:

\[
x - 4x = 9
\]

\[
-3x = 9
\]

### Step 5: Solve for \(x\)
Divide both sides by \(-3\):

\[
x = \frac{9}{-3} = -3
\]

### Final Answer:
\[
x = -3
\]

If you have any questions or want further details, feel free to ask!

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Here are 5 related questions:
1. How do you find the least common denominator for multiple terms?
2. What are the properties of rational expressions when multiplied by their least common denominator?
3. How do you handle combining like terms in an equation with variables on both sides?
4. What other techniques can be used to eliminate fractions from equations?
5. How does factoring help in solving quadratic equations?

**Tip**: Always check your solution by plugging the value of \(x\) back into the original equation to ensure both sides are equal.

Solve: 1/6x = 2x + 5/3x² - 1/6x². If final answer is a fraction, write as a simplified fraction, #/#. NO DECIMALS. X =

Learn how to solve the rational equation 1/6x = 2x + 5/3x² - 1/6x² step-by-step, with key algebraic techniques like multiplying by the least common denominator and combining like terms. This problem involves simplifying fractional expressions and isolating the variable x. Suitable for students in Grades 8-10.

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