Given f(x) = (3x − 2)/(5x + 4) and g(x) = (7x + 2)/(4x - 3), find (a) f⁻¹(x), (b) g⁻¹(x), and (c) the values of x if g⁻¹(x) = 1 − f⁻¹(x).

Let $y = f(x)$, so $y = \frac{3x - 2}{5x + 4}$.
To find the inverse function, swap $x$ and $y$:

$$x = \frac{3y - 2}{5y + 4}$$

Multiply both sides by $(5y + 4)$:

$$x(5y + 4) = 3y - 2$$ $$5xy + 4x = 3y - 2$$

Rearrange the terms to isolate $y$:

$$5xy - 3y = -2 - 4x$$

Factor out $y$:

$$y(5x - 3) = -2 - 4x$$

Solve for $y$:

$$y = \frac{-2 - 4x}{5x - 3}$$

This can be written as:

$$f^{-1}(x) = \boxed{\mathbf{\frac{4x + 2}{3 - 5x}}}$$

Let $y = g(x)$, so $y = \frac{7x + 2}{4x - 3}$.
To find the inverse function, swap $x$ and $y$:

$$x = \frac{7y + 2}{4y - 3}$$

Multiply both sides by $(4y - 3)$:

$$x(4y - 3) = 7y + 2$$ $$4xy - 3x = 7y + 2$$

Rearrange the terms to isolate $y$:

$$4xy - 7y = 2 + 3x$$

Factor out $y$:

$$y(4x - 7) = 2 + 3x$$

Solve for $y$:

$$y = \frac{2 + 3x}{4x - 7}$$

So,

$$g^{-1}(x) = \boxed{\mathbf{\frac{3x + 2}{4x - 7}}}$$

Substitute the expressions for $f^{-1}(x)$ and $g^{-1}(x)$ into the equation:

$$\frac{3x + 2}{4x - 7} = 1 - \frac{4x + 2}{3 - 5x}$$

First, simplify the right side:

$$1 - \frac{4x + 2}{3 - 5x} = \frac{3 - 5x}{3 - 5x} - \frac{4x + 2}{3 - 5x} = \frac{(3 - 5x) - (4x + 2)}{3 - 5x} = \frac{3 - 5x - 4x - 2}{3 - 5x} = \frac{1 - 9x}{3 - 5x}$$

Now the equation is:

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

Cross-multiply:

$$(3x + 2)(3 - 5x) = (1 - 9x)(4x - 7)$$

Expand both sides:

$$9x - 15x^2 + 6 - 10x = 4x - 7 - 36x^2 + 63x$$ $$-15x^2 - x + 6 = -36x^2 + 67x - 7$$

Move all terms to one side to form a quadratic equation:

$$-15x^2 + 36x^2 - x - 67x + 6 + 7 = 0$$ $$21x^2 - 68x + 13 = 0$$

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=21$, $b=-68$, $c=13$:

$$x = \frac{-(-68) \pm \sqrt{(-68)^2 - 4(21)(13)}}{2(21)}$$ $$x = \frac{68 \pm \sqrt{4624 - 1092}}{42}$$ $$x = \frac{68 \pm \sqrt{3532}}{42}$$

Calculate the square root:

$$\sqrt{3532} \approx 59.43063$$

Now find the two values for $x$:

$$x_1 = \frac{68 + 59.43063}{42} = \frac{127.43063}{42} \approx 3.03406$$ $$x_2 = \frac{68 - 59.43063}{42} = \frac{8.56937}{42} \approx 0.20398$$

Rounding to two decimal places:

$$x_1 \approx 3.03$$ $$x_2 \approx 0.20$$

The values of $x$ are $\boxed{\mathbf{3.03 \text{ and } 0.20}}$.