To express [tex]\(\frac{1}{9x^2 - 25} - \frac{1}{6x + 10}\)[/tex] as a single fraction in its simplest form, we need to follow a few steps.
1. Find the denominators and factorize them if possible:
- The first denominator is [tex]\(9x^2 - 25\)[/tex]. This is a difference of squares and can be factorized as follows:
[tex]\[
9x^2 - 25 = (3x - 5)(3x + 5)
\][/tex]
- The second denominator is [tex]\(6x + 10\)[/tex]. This can be simplified by factoring out the common factor of 2:
[tex]\[
6x + 10 = 2(3x + 5)
\][/tex]
2. Identify the least common denominator (LCD):
- The least common denominator will be the product of the distinct factors from both denominators. Thus, the LCD is:
[tex]\[
2(3x - 5)(3x + 5)
\][/tex]
3. Rewrite each fraction with the LCD as the common denominator:
For [tex]\(\frac{1}{9x^2 - 25}\)[/tex], we need to express it with the LCD:
[tex]\[
\frac{1}{9x^2 - 25} = \frac{1}{(3x-5)(3x+5)}
\][/tex]
To have the LCD, it needs to be multiplied by 2:
[tex]\[
\frac{1}{(3x-5)(3x+5)} = \frac{2}{2(3x-5)(3x+5)}
\][/tex]
For [tex]\(\frac{1}{6x + 10}\)[/tex], we need to express it with the LCD:
[tex]\[
\frac{1}{6x + 10} = \frac{1}{2(3x+5)}
\][/tex]
To have the LCD, it needs to be multiplied by [tex]\(3x-5\)[/tex]:
[tex]\[
\frac{1}{2(3x+5)} = \frac{3x - 5}{2(3x-5)(3x+5)}
\][/tex]
4. Combine the fractions:
Now, we subtract the two fractions:
[tex]\[
\frac{2}{2(3x-5)(3x+5)} - \frac{3x - 5}{2(3x-5)(3x+5)}
\][/tex]
Since both fractions have the same denominator, we can combine them over the common denominator:
[tex]\[
\frac{2 - (3x - 5)}{2(3x-5)(3x+5)}
\][/tex]
5. Simplify the numerator:
Simplify the expression in the numerator:
[tex]\[
2 - (3x - 5) = 2 - 3x + 5 = 7 - 3x
\][/tex]
Therefore, the combined fraction becomes:
[tex]\[
\frac{7 - 3x}{2(3x-5)(3x+5)}
\][/tex]
Thus, the expression [tex]\(\frac{1}{9x^2 - 25} - \frac{1}{6x + 10}\)[/tex] simplifies to:
[tex]\[
\boxed{\frac{7 - 3x}{2(9x^2 - 25)}}
\][/tex]
