Let's break down the mathematical steps in the image:

Problem Interpretation

The image shows the following expression and equation:

1. $\frac{2x + 12}{3}$
2. $\frac{4x}{9} + \frac{2x}{3} = \frac{5}{7} - \frac{7}{6}$

The first part $\frac{2x + 12}{3}$ looks like an expression that may relate to the equation below it, but we'll focus on solving the main equation provided:

$\frac{4x}{9} + \frac{2x}{3} = \frac{5}{7} - \frac{7}{6}$

Step-by-Step Solution

Step 1: Simplify each side of the equation

1. Left Side: $\frac{4x}{9} + \frac{2x}{3}$

   • To add these terms, find a common denominator. The common denominator for $9$ and $3$ is $9$.
   • Rewrite $\frac{2x}{3}$ as $\frac{6x}{9}$.
   • So, $\frac{4x}{9} + \frac{2x}{3} = \frac{4x + 6x}{9} = \frac{10x}{9}$.

   Thus, the left side simplifies to $\frac{10x}{9}$.

2. Right Side: $\frac{5}{7} - \frac{7}{6}$

   • To subtract these fractions, find a common denominator. The common denominator for $7$ and $6$ is $42$.
   • Rewrite $\frac{5}{7}$ as $\frac{30}{42}$ and $\frac{7}{6}$ as $\frac{49}{42}$.
   • So, $\frac{5}{7} - \frac{7}{6} = \frac{30 - 49}{42} = \frac{-19}{42}$.

   Thus, the right side simplifies to $\frac{-19}{42}$.

Step 2: Set up the simplified equation

After simplifying both sides, we get: $\frac{10x}{9} = \frac{-19}{42}$

Step 3: Solve for $x$

To isolate $x$, cross-multiply to eliminate the fractions: $10x \cdot 42 = -19 \cdot 9$ $420x = -171$ Now, divide by $420$ to solve for $x$: $x = \frac{-171}{420}$ Simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is $3$: $x = \frac{-57}{140}$

Final Answer

$x = \frac{-57}{140}$

Would you like further details on any specific step, or do you have additional questions?

Here are five related questions you might find useful:

1. How can I verify the solution by substituting the value of $x$ back into the original equation?
2. What are common methods to find the least common multiple (LCM) when simplifying fractions?
3. How can I solve equations that involve more complex fractions?
4. What are some tips for simplifying expressions before solving an equation?
5. How would the process change if there were more terms with different variables?

Tip: Always try to simplify fractions and expressions as much as possible before attempting to solve equations. This can make calculations easier and reduce the chance of error.