https://file.aiquickdraw.com/mathbot/file/1728798121745esx7p0gx.jpg

The equation provided in the image is:

\[
\frac{2x - 9}{x^2} - \frac{3}{5x} = \frac{5}{x}
\]

Let's solve this step by step.

### Step 1: Eliminate the denominators
To clear the denominators, we can multiply through by the least common denominator (LCD). The denominators are \(x^2\), \(5x\), and \(x\). The LCD is \(5x^2\).

Multiplying each term by \(5x^2\):

\[
5x^2 \cdot \left( \frac{2x - 9}{x^2} \right) - 5x^2 \cdot \left( \frac{3}{5x} \right) = 5x^2 \cdot \left( \frac{5}{x} \right)
\]

This simplifies as follows:
- First term: \(5x^2 \cdot \frac{2x - 9}{x^2} = 5(2x - 9)\)
- Second term: \(5x^2 \cdot \frac{3}{5x} = x \cdot 3 = 3x\)
- Third term: \(5x^2 \cdot \frac{5}{x} = 25x\)

So, the equation becomes:

\[
5(2x - 9) - 3x = 25x
\]

### Step 2: Expand and simplify
Expand the terms:

\[
10x - 45 - 3x = 25x
\]

Combine like terms:

\[
(10x - 3x) - 45 = 25x
\]
\[
7x - 45 = 25x
\]

### Step 3: Isolate \(x\)
Now, move all terms involving \(x\) to one side:

\[
7x - 25x = 45
\]
\[
-18x = 45
\]

### Step 4: Solve for \(x\)
Divide both sides by \(-18\):

\[
x = \frac{45}{-18} = -\frac{5}{2}
\]

### Final Answer:
The solution is:

\[
x = -\frac{5}{2}
\]

Would you like a further breakdown or more details on this solution?

Here are 5 related questions for deeper practice:
1. What is the least common denominator of \(\frac{1}{x}\) and \(\frac{2}{x^2}\)?
2. How would you solve \(\frac{3x - 2}{x^2} = \frac{1}{x}\)?
3. Can you factor the expression \(2x^2 - 7x + 3 = 0\)?
4. How do you multiply and simplify \(\frac{x-3}{x+2} \times \frac{x+4}{x-1}\)?
5. How would you graph \(y = \frac{5}{x} - \frac{3}{x+1}\)?

**Tip**: Always check your solution by substituting the value of \(x\) back into the original equation to ensure it satisfies the equation.

Solve the equation: (2x - 9)/x^2 - 3/(5x) = 5/x.

Learn how to solve the rational equation (2x - 9)/x^2 - 3/(5x) = 5/x using step-by-step algebraic methods. This problem covers key concepts like the Least Common Denominator and rational expression manipulation. Suitable for students in Grades 9-11.

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