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We are given the function \( g(x) = \frac{7}{8}x - \frac{6}{7} \), and we need to find the value of \( x \) that corresponds to \( g(x) = \frac{5}{6} \).

Let's set up the equation:

\[
\frac{7}{8}x - \frac{6}{7} = \frac{5}{6}
\]

To solve for \( x \), we first add \( \frac{6}{7} \) to both sides:

\[
\frac{7}{8}x = \frac{5}{6} + \frac{6}{7}
\]

We now need to find a common denominator to add the fractions on the right-hand side. The least common denominator (LCD) of 6 and 7 is 42.

Convert both fractions to have a denominator of 42:

\[
\frac{5}{6} = \frac{35}{42}, \quad \frac{6}{7} = \frac{36}{42}
\]

Now, add them:

\[
\frac{35}{42} + \frac{36}{42} = \frac{71}{42}
\]

So, the equation becomes:

\[
\frac{7}{8}x = \frac{71}{42}
\]

Next, multiply both sides by the reciprocal of \( \frac{7}{8} \), which is \( \frac{8}{7} \), to isolate \( x \):

\[
x = \frac{8}{7} \times \frac{71}{42}
\]

Simplify the multiplication:

\[
x = \frac{8 \times 71}{7 \times 42} = \frac{568}{294}
\]

Now simplify the fraction by dividing both the numerator and the denominator by 2:

\[
x = \frac{284}{147}
\]

Thus, the value of \( x \) is \( \frac{284}{147} \).

---

Would you like more details on any part of this process or have any questions? Here are 5 related questions you might find useful:

1. How do you determine the least common denominator between two fractions?
2. Can you explain more about multiplying by the reciprocal?
3. How can I verify the solution by plugging the value of \( x \) back into the equation?
4. What are other methods to solve linear equations involving fractions?
5. What strategies can simplify the addition of fractions with different denominators?

**Tip:** Always simplify fractions step by step, especially in problems involving multiple fractions. This helps avoid errors and makes the process clearer.

Let g(x) = (7/8)x - (6/7) and find the value of x that corresponds to g(x) = 5/6.

This detailed solution shows how to solve a linear equation involving fractions where g(x) = (7/8)x - (6/7), and find x when g(x) = 5/6. Learn how to handle fraction addition, common denominators, and solving linear equations step-by-step.

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