X/2+3x - 5/3x-2 = 15x+10/4-9x

To solve the equation

\[ \frac{x}{2} + 3x - \frac{5}{3}x - 2 = \frac{15x + 10}{4 - 9x}, \]

let's first simplify both sides step by step.

Step 1: Simplify the left-hand side

Combine the terms on the left side:

\[ \frac{x}{2} + 3x - \frac{5}{3}x - 2. \]

To combine these terms, it's often easier to get a common denominator for the fractional terms. The common denominator among the fractions can be 6.

Convert \(\frac{x}{2}\) and \(\frac{5}{3}x\):

\[ \frac{x}{2} = \frac{3x}{6}, \quad \text{and} \quad \frac{5}{3}x = \frac{10x}{6}. \]

Now rewrite the left-hand side:

\[ \frac{3x}{6} + 3x - \frac{10x}{6} - 2. \]

Note that \(3x\) can also be rewritten as \(\frac{18x}{6}\):

\[ = \frac{3x}{6} + \frac{18x}{6} - \frac{10x}{6} - 2 = \frac{3x + 18x - 10x}{6} - 2. \]

Combine the \(x\) terms:

\[ = \frac{11x}{6} - 2. \]

Convert \(-2\) to sixths for simplicity:

\[ -2 = -\frac{12}{6}. \]

Thus, the left-hand side simplifies to:

\[ \frac{11x - 12}{6}. \]

Step 2: Simplify the right-hand side

The right-hand side is:

\[ \frac{15x + 10}{4 - 9x}. \]

This term does not simplify further directly, so we will keep it as it is for now:

\[ \frac{15x + 10}{4 - 9x}. \]

Step 3: Set the left-hand side equal to the right-hand side

Now we have:

\[ \frac{11x - 12}{6} = \frac{15x + 10}{4 - 9x}. \]

Step 4: Cross-multiply

Cross-multiply to eliminate the fractions. This gives us:

\[ (11x - 12)(4 - 9x) = (15x + 10)(6). \]

Step 5: Expand both sides

Left side:

\[ 11x \cdot 4 - 11x \cdot 9x - 12 \cdot 4 + 12 \cdot 9x = 44x - 99x^2 - 48 + 108x. \]

Combine like terms:

\[ -99x^2 + (44x + 108x) - 48 = -99x^2 + 152x - 48. \]

Right side:

\[ 15x \cdot 6 + 10 \cdot 6 = 90x + 60. \]

Step 6: Set the entire equation to zero

Putting both sides together:

\[ -99x^2 + 152x - 48 = 90x + 60. \]

Subtract \(90x + 60\) from both sides:

\[ -99x^2 + 152x - 90x - 48 - 60 = 0. \]

This simplifies to:

\[ -99x^2 + 62x - 108 = 0. \]

Step 7: Multiply by -1 to make calculations easier

\[ 99x^2 - 62x + 108 = 0. \]

Step 8: Solve using the quadratic formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

Here \(a = 99\), \(b = -62\), and \(c = 108\).

1. Calculate the discriminant \(b^2 - 4ac\):

\[ (-62)^2 - 4 \cdot 99 \cdot 108 = 3844 - 42792 = -38948. \]

Since the discriminant is negative, there are no real roots.

Conclusion:

The equation has no real solutions.