Solve the equation 3sqrt(4x−3) − sqrt(5x+1) = 5 for integer values of x.

For the square roots to be defined, the expressions under them must be non-negative:

$$4x-3 \ge 0 \implies 4x \ge 3 \implies x \ge \frac{3}{4}$$ $$5x+1 \ge 0 \implies 5x \ge -1 \implies x \ge -\frac{1}{5}$$

Combining these conditions, we must have $x \ge \frac{3}{4}$. Since $x$ is specified as an integer, $x$ must be $1, 2, 3, \dots$.

The given equation is $3\sqrt{4x-3} - \sqrt{5x+1} = 5$.
Move the second radical term to the right side of the equation:

$$3\sqrt{4x-3} = 5 + \sqrt{5x+1}$$

$$(3\sqrt{4x-3})^2 = (5 + \sqrt{5x+1})^2$$ $$9(4x-3) = 5^2 + 2(5)\sqrt{5x+1} + (\sqrt{5x+1})^2$$ $$36x - 27 = 25 + 10\sqrt{5x+1} + 5x+1$$ $$36x - 27 = 26 + 5x + 10\sqrt{5x+1}$$

Subtract 26 and $5x$ from both sides:

$$36x - 5x - 27 - 26 = 10\sqrt{5x+1}$$ $$31x - 53 = 10\sqrt{5x+1}$$

$$(31x - 53)^2 = (10\sqrt{5x+1})^2$$ $$(31x)^2 - 2(31x)(53) + 53^2 = 100(5x+1)$$ $$961x^2 - 3286x + 2809 = 500x + 100$$

$$961x^2 - 3286x - 500x + 2809 - 100 = 0$$ $$961x^2 - 3786x + 2709 = 0$$

Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$$x = \frac{-(-3786) \pm \sqrt{(-3786)^2 - 4(961)(2709)}}{2(961)}$$ $$x = \frac{3786 \pm \sqrt{14333796 - 10413396}}{1922}$$ $$x = \frac{3786 \pm \sqrt{3920400}}{1922}$$

Calculate the square root: $\sqrt{3920400} = 1980$.

$$x = \frac{3786 \pm 1980}{1922}$$

This gives two possible solutions:

$$x_1 = \frac{3786 + 1980}{1922} = \frac{5766}{1922} = 3$$ $$x_2 = \frac{3786 - 1980}{1922} = \frac{1806}{1922} = \frac{903}{961}$$

We established that $x$ must be an integer and $x \ge \frac{3}{4}$.
For $x_1 = 3$: This is an integer and $3 \ge \frac{3}{4}$.
Substitute $x=3$ into the original equation:
$3\sqrt{4(3)-3} - \sqrt{5(3)+1} = 3\sqrt{12-3} - \sqrt{15+1}$
$= 3\sqrt{9} - \sqrt{16}$
$= 3(3) - 4$
$= 9 - 4 = 5$
This matches the right side of the equation, so $x=3$ is a valid solution.

For $x_2 = \frac{903}{961}$: This is not an integer, so it is not a valid solution according to the problem's condition that $x$ is an integer.

The only integer solution is $x=3$.