https://file.aiquickdraw.com/mathbot/file/1726184505787h1s3fe09.jpg

You've provided a photo of a rational equation to solve. Let's go through the solution step-by-step.

The equation given is:
\[ \frac{3x - 1}{5} + \frac{2x - 1}{6} = \frac{5}{6} \]

First, we need to find a common denominator for the fractions on the left side, which is 30. We can rewrite each fraction with this common denominator:
\[ \frac{3x - 1}{5} = \frac{6(3x - 1)}{30} \]
\[ \frac{2x - 1}{6} = \frac{5(2x - 1)}{30} \]
\[ \frac{5}{6} = \frac{25}{30} \]

Now, rewrite the equation:
\[ \frac{18x - 6 + 10x - 5}{30} = \frac{25}{30} \]

Combine like terms in the numerator:
\[ \frac{28x - 11}{30} = \frac{25}{30} \]

Since the denominators are the same, we can equate the numerators:
\[ 28x - 11 = 25 \]

Now, solve for \(x\):
\[ 28x = 25 + 11 \]
\[ 28x = 36 \]
\[ x = \frac{36}{28} \]
\[ x = \frac{9}{7} \]

Thus, \(x = \frac{9}{7}\).

Do you want to explore further details or do you have any questions about the steps? Here are some questions you might consider:

1. How can you check if \(\frac{9}{7}\) is indeed a correct solution to the equation?
2. What happens if you were to substitute different values for \(x\) into the original equation?
3. What would be the graphical interpretation of this equation?
4. How would the solution change if the equation on the right was different, say \(\frac{1}{2}\)?
5. Are there scenarios where a rational equation like this would have no solution or infinite solutions?

**Tip:** Always simplify the fractions during your calculation to make calculations easier and avoid mistakes.

Solve the rational equations given in the image: (1) 2/(3 - 5) = 3x, (2) (2 + 1)/(6 - 2x) = 3, and (3) (4 + 1)/4 = 5.

Learn how to solve rational equations with step-by-step solutions, involving operations with fractions, equating terms, and solving linear equations. Ideal for students in Grades 9-12 looking to strengthen their understanding of algebraic concepts.

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