To simplify the expression [tex]\(\frac{6 - 5x}{-3x}\)[/tex], follow these steps:
1. Recognize the structure of the expression:
The expression is a rational function in terms of [tex]\(x\)[/tex], having a numerator [tex]\(6 - 5x\)[/tex] and a denominator [tex]\(-3x\)[/tex].
2. Rewrite the negative sign:
The negative sign in the denominator affects the entire fraction, so you can factor it out:
[tex]\[
\frac{6 - 5x}{-3x} = -\frac{6 - 5x}{3x}
\][/tex]
3. Distribute the negative sign:
Distribute the negative sign to each term in the numerator:
[tex]\[
-\frac{6 - 5x}{3x} = -\left(\frac{6}{3x} - \frac{5x}{3x}\right)
\][/tex]
4. Simplify each term separately:
- For [tex]\(\frac{6}{3x}\)[/tex]:
[tex]\[
\frac{6}{3x} = \frac{6}{3} \cdot \frac{1}{x} = 2 \cdot \frac{1}{x} = \frac{2}{x}
\][/tex]
- For [tex]\(\frac{5x}{3x}\)[/tex]:
[tex]\[
\frac{5x}{3x} = \frac{5}{3} \cdot \frac{x}{x} = \frac{5}{3} \cdot 1 = \frac{5}{3}
\][/tex]
5. Combine the simplified terms:
Now we combine the results, keeping in mind the redistributed negative sign:
[tex]\[
-\left(\frac{2}{x} - \frac{5}{3}\right) = -\frac{2}{x} + \frac{5}{3}
\][/tex]
6. Arrange the terms:
Rearrange the simplified terms for clarity:
[tex]\[
\frac{5}{3} - \frac{2}{x}
\][/tex]
Thus, the original expression [tex]\(\frac{6 - 5x}{-3x}\)[/tex] simplifies to:
[tex]\[
\frac{5}{3} - \frac{2}{x}
\][/tex]
