Find $x + \dfrac 1x$ if\[3x+\frac{2}{x}-4=2(x+9)-\left(7-\frac{1}{x}\right).\]

We are tasked with finding $ x + \frac{1}{x} $ given the equation:

\[
3x + \frac{2}{x} - 4 = 2(x + 9) - \left(7 - \frac{1}{x}\right).
\]

### Step 1: Simplify the right-hand side

First, distribute on the right-hand side:

\[
2(x + 9) = 2x + 18.
\]

Now substitute back into the equation:

\[
3x + \frac{2}{x} - 4 = 2x + 18 - \left(7 - \frac{1}{x}\right).
\]

Simplify the expression inside the parentheses on the right:

\[
7 - \frac{1}{x}.
\]

This gives:

\[
3x + \frac{2}{x} - 4 = 2x + 18 - 7 + \frac{1}{x}.
\]

Simplify further:

\[
3x + \frac{2}{x} - 4 = 2x + 11 + \frac{1}{x}.
\]

### Step 2: Move all terms involving $ x $ and $ \frac{1}{x} $ to one side

Subtract $ 2x + \frac{1}{x} $ from both sides:

\[
(3x - 2x) + \left(\frac{2}{x} - \frac{1}{x}\right) - 4 = 11.
\]

This simplifies to:

\[
x + \frac{1}{x} - 4 = 11.
\]

### Step 3: Solve for $ x + \frac{1}{x} $

Add 4 to both sides:

\[
x + \frac{1}{x} = 15.
\]

Thus, $ x + \frac{1}{x} = 15 $.

Would you like further explanation or details?

### Related questions:
1. How do you solve for $ x $ if $ x + \frac{1}{x} = 15 $?
2. What happens when $ x + \frac{1}{x} = 0 $?
3. How can you express $ x^2 + \frac{1}{x^2} $ in terms of $ x + \frac{1}{x} $?
4. How would you handle fractional equations involving higher degrees of $ x $?
5. Can you solve for other expressions like $ x^3 + \frac{1}{x^3} $ based on this result?

**Tip:** When dealing with rational expressions, try to simplify both sides of the equation to move terms involving $ x $ and $ \frac{1}{x} $ together.

Find $x + \dfrac{1}{x}$ if \[3x + \frac{2}{x} - 4 = 2(x + 9) - \left(7 - \frac{1}{x}\right).\]

Learn how to solve the rational expression problem 3x + 2/x - 4 = 2(x + 9) - (7 - 1/x) and find x + 1/x. This detailed solution involves simplifying rational terms and isolating variables, making it ideal for students in Grades 9-12.

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