\frac{x-\frac{5+y}{4}}{\frac{3}{x}-\frac{2}{y}}

The expression you've provided is:

\[
\frac{x - \frac{5 + y}{4}}{\frac{3}{x} - \frac{2}{y}}
\]

To simplify this expression, let's proceed step by step.

### Step 1: Simplify the numerator
The numerator is:
\[
x - \frac{5 + y}{4}
\]
To combine these terms, let's rewrite \(x\) as \(\frac{4x}{4}\):
\[
\frac{4x}{4} - \frac{5 + y}{4} = \frac{4x - (5 + y)}{4} = \frac{4x - 5 - y}{4}
\]

### Step 2: Simplify the denominator
The denominator is:
\[
\frac{3}{x} - \frac{2}{y}
\]
To combine these fractions, we need a common denominator, which would be \(xy\):
\[
\frac{3y - 2x}{xy}
\]

### Step 3: Combine the simplified numerator and denominator
Now, we can write the entire expression as:
\[
\frac{\frac{4x - 5 - y}{4}}{\frac{3y - 2x}{xy}}
\]

To simplify this further, multiply the numerator by the reciprocal of the denominator:
\[
\frac{4x - 5 - y}{4} \times \frac{xy}{3y - 2x} = \frac{(4x - 5 - y) \cdot xy}{4 \cdot (3y - 2x)}
\]

Finally, simplify the expression:
\[
\frac{xy(4x - 5 - y)}{4(3y - 2x)}
\]

This is the simplified form of the given expression.

Would you like further details or have any questions?

Here are some related questions to explore further:

1. How can the expression be simplified if specific values are given for \(x\) and \(y\)?
2. How does the simplification change if \(x = y\)?
3. What are the potential restrictions on \(x\) and \(y\) for the expression to be defined?
4. How can the expression be interpreted geometrically?
5. What happens if the numerator or denominator equals zero?

**Tip:** When simplifying complex fractions, always look for common denominators to combine terms effectively.

Learn how to simplify the fraction expression (x - (5 + y)/4) / ((3/x) - (2/y)) step-by-step with a detailed solution. This problem covers fraction simplification in algebra, involving variables x and y. Suitable for students in Grades 9-12.

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