The problem is to solve for $x$ in the following equation: $\frac{4x-3}{5} + \frac{5x+9}{4} = \frac{2x}{3} + \frac{15}{2} - \frac{7x}{6}$.

1. Problem Description

The problem is to solve for $x$ in the following equation:

$\frac{4x-3}{5} + \frac{5x+9}{4} = \frac{2x}{3} + \frac{15}{2} - \frac{7x}{6}$.

2. Solution Steps

First, multiply both sides of the equation by the least common multiple of the denominators, which is $LCM(5, 4, 3, 2, 6) = 60$:

$60 \left( \frac{4x-3}{5} + \frac{5x+9}{4} \right) = 60 \left( \frac{2x}{3} + \frac{15}{2} - \frac{7x}{6} \right)$

$12(4x-3) + 15(5x+9) = 20(2x) + 30(15) - 10(7x)$

$48x - 36 + 75x + 135 = 40x + 450 - 70x$

Combine like terms on each side of the equation:

$(48x + 75x) + (-36 + 135) = (40x - 70x) + 450$

$123x + 99 = -30x + 450$

Add $30x$ to both sides:

$123x + 30x + 99 = -30x + 30x + 450$

$153x + 99 = 450$

Subtract $99$ from both sides:

$153x + 99 - 99 = 450 - 99$

$153x = 351$

Divide both sides by $153$:

$x = \frac{351}{153}$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is $9$:

$x = \frac{351 \div 9}{153 \div 9} = \frac{39}{17}$

3. Final Answer

$x = \frac{39}{17}$