Calculate the sum and difference of these fractions: 3/4 + (-4/3) + 5/6, and verify if it equals -4/3 + 5/6 + 3/4.

The equation is:

$$\frac{3}{4} + \frac{(-4)}{3} + \frac{5}{6} = \frac{(-4)}{3} + \frac{5}{6} + \frac{3}{4}$$

Simplify the signs:

$$\frac{3}{4} - \frac{4}{3} + \frac{5}{6} = -\frac{4}{3} + \frac{5}{6} + \frac{3}{4}$$

To add and subtract fractions, find a common denominator. The denominators are 4, 3, and 6. The least common multiple (LCM) of 4, 3, and 6 is 12.
Convert each fraction to have a denominator of 12:

$$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$$ $$-\frac{4}{3} = -\frac{4 \times 4}{3 \times 4} = -\frac{16}{12}$$ $$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$$

Now, add these fractions:

$$\text{LHS} = \frac{9}{12} - \frac{16}{12} + \frac{10}{12} = \frac{9 - 16 + 10}{12}$$ $$\text{LHS} = \frac{-7 + 10}{12} = \frac{3}{12}$$

Simplify the fraction:

$$\text{LHS} = \frac{1}{4}$$

Using the same common denominator (12) for the fractions on the RHS:

$$-\frac{4}{3} = -\frac{16}{12}$$ $$\frac{5}{6} = \frac{10}{12}$$ $$\frac{3}{4} = \frac{9}{12}$$

Now, add these fractions:

$$\text{RHS} = -\frac{16}{12} + \frac{10}{12} + \frac{9}{12} = \frac{-16 + 10 + 9}{12}$$ $$\text{RHS} = \frac{-6 + 9}{12} = \frac{3}{12}$$

Simplify the fraction:

$$\text{RHS} = \frac{1}{4}$$

Since LHS $= \frac{1}{4}$ and RHS $= \frac{1}{4}$, both sides are equal.
Therefore, the given equation is true.

$$\frac{1}{4} = \frac{1}{4}$$