To simplify the expression:

\[
\frac{7}{36 - x^2} - \frac{3}{6 + x}
\]

### Step 1: Factorize the denominator \(36 - x^2\).
The denominator \(36 - x^2\) is a difference of squares, which can be factored as:

\[
36 - x^2 = (6 - x)(6 + x)
\]

So, the expression becomes:

\[
\frac{7}{(6 - x)(6 + x)} - \frac{3}{6 + x}
\]

### Step 2: Find the common denominator.
The common denominator between \((6 - x)(6 + x)\) and \(6 + x\) is \((6 - x)(6 + x)\).

We need to rewrite the second fraction so that both fractions have the same denominator:

\[
\frac{3}{6 + x} = \frac{3(6 - x)}{(6 - x)(6 + x)}
\]

Thus, the expression becomes:

\[
\frac{7}{(6 - x)(6 + x)} - \frac{3(6 - x)}{(6 - x)(6 + x)}
\]

### Step 3: Combine the fractions.
Now that the fractions have the same denominator, we can combine them:

\[
\frac{7 - 3(6 - x)}{(6 - x)(6 + x)}
\]

### Step 4: Simplify the numerator.
Distribute the \(3\) in the numerator:

\[
7 - 3(6 - x) = 7 - 18 + 3x = 3x - 11
\]

### Step 5: Write the final simplified expression.
So, the simplified expression is:

\[
\frac{3x - 11}{(6 - x)(6 + x)}
\]

This is the fully simplified form.

---

Do you have any questions or would you like further details? Here are some related problems for practice:

1. Simplify \(\frac{5}{25 - x^2} - \frac{2}{5 + x}\).
2. Simplify \(\frac{4}{16 - y^2} + \frac{3}{4 - y}\).
3. Simplify \(\frac{9}{49 - z^2} - \frac{4}{7 - z}\).
4. Factor \(49 - a^2\) and simplify \(\frac{7}{49 - a^2} - \frac{1}{7 - a}\).
5. Simplify \(\frac{10}{x^2 - 4} - \frac{5}{x - 2}\).

**Tip:** When simplifying rational expressions, always look for common factors or patterns like the difference of squares to make simplifications easier.

Learn how to fully simplify the rational expression 7/(36-x^2) - 3/(6+x) by factoring using the difference of squares and combining fractions with a common denominator. This step-by-step solution covers key algebra concepts suitable for students in Grades 8-10.

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