The given expression is:

$\frac{4}{9} - \frac{5}{6} + \frac{1}{4} - \frac{9}{4} + \frac{4}{1}$

Let's break this down step by step:

1. Combine like terms: First, let's simplify the expression by combining terms with common denominators if possible. We need to handle fractions with different denominators.

$\frac{4}{9} - \frac{5}{6} + \frac{1}{4} - \frac{9}{4} + \frac{4}{1}$

2. Find a common denominator for all fractions:

   The denominators are 9, 6, 4, and 1. The least common denominator (LCD) of these numbers is 36 (since 36 is the smallest number that 9, 6, 4, and 1 divide into).

3. Rewrite each fraction with denominator 36:

$\frac{4}{9} = \frac{4 \times 4}{9 \times 4} = \frac{16}{36}$ $\frac{5}{6} = \frac{5 \times 6}{6 \times 6} = \frac{30}{36}$ $\frac{1}{4} = \frac{1 \times 9}{4 \times 9} = \frac{9}{36}$ $\frac{9}{4} = \frac{9 \times 9}{4 \times 9} = \frac{81}{36}$ $\frac{4}{1} = \frac{4 \times 36}{1 \times 36} = \frac{144}{36}$

4. Now, substitute back into the expression:

$\frac{16}{36} - \frac{30}{36} + \frac{9}{36} - \frac{81}{36} + \frac{144}{36}$

5. Combine the fractions:

$\frac{16 - 30 + 9 - 81 + 144}{36} = \frac{58}{36}$

6. Simplify the fraction:

$\frac{58}{36} = \frac{29}{18}$

So, the value of the expression is:

$\boxed{\frac{29}{18}}$

Would you like more details on the steps or have any further questions? Here are some additional questions for practice:

1. How do you find the least common denominator (LCD) of fractions with different denominators?
2. How do you simplify a fraction after combining terms?
3. What is the process for subtracting fractions with different denominators?
4. How do you convert an improper fraction to a mixed number?
5. How do you add fractions with mixed denominators?

Tip: When dealing with fractions, always ensure you work with the least common denominator (LCD) for easier calculations!