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Find a common denominator. show solutin step by step

To find a common denominator for the expression:

\[
\frac{1}{4 - X} - \frac{3}{X(X - 4)}
\]

we'll follow these steps:

### Step 1: Identify the Denominators
The denominators of the given fractions are:
1. \( 4 - X \)
2. \( X(X - 4) \)

### Step 2: Simplify the Denominators
Notice that \( 4 - X \) can be rewritten as \( -(X - 4) \). This allows us to match it with the term in the second denominator, \( (X - 4) \).

So, the denominators become:
1. \( -(X - 4) \)
2. \( X(X - 4) \)

### Step 3: Find the Common Denominator
The common denominator is the product of all unique factors appearing in the denominators. Here, the common denominator is:

\[
-X(X - 4)
\]

### Step 4: Rewrite Each Fraction with the Common Denominator
1. **First Fraction:**
   \[
   \frac{1}{4 - X} = \frac{-1}{X - 4}
   \]

   Now, multiply the numerator and denominator by \( X \) to match the common denominator:

   \[
   \frac{-1}{X - 4} = \frac{-X}{-X(X - 4)} = \frac{-X}{X(X - 4)}
   \]

2. **Second Fraction:**
   This fraction already has the common denominator:

   \[
   \frac{3}{X(X - 4)}
   \]

### Step 5: Combine the Fractions
Now, we can combine the two fractions over the common denominator:

\[
\frac{-X}{X(X - 4)} - \frac{3}{X(X - 4)} = \frac{-X - 3}{X(X - 4)}
\]

So, the simplified form of the given expression with a common denominator is:

\[
\frac{-X - 3}{X(X - 4)}
\]

Would you like a more detailed explanation, or do you have any other questions related to this problem?

### Related Questions:
1. How do you find the least common denominator for multiple fractions?
2. What are the steps to simplify complex rational expressions?
3. How do you handle negative signs in denominators when finding common denominators?
4. What are the restrictions on \( X \) for the given expression?
5. How can you check if the simplified expression is correct?

### Tip:
When working with rational expressions, always factor and simplify denominators first to make finding a common denominator easier!

Find a common denominator. Show solution step by step.

Learn how to find the common denominator for the rational expression 1/(4 - X) - 3/X(X - 4). Follow a step-by-step solution that includes identifying denominators, simplifying them, and combining fractions over a common denominator. Suitable for high school students in Grades 9-12.

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