Begin by simplifying the numerator.

$\displaystyle \frac{1}{4}+\frac{5}{7}$ has a common denominator of $\displaystyle 28$.  Therefore, we can rewrite it as:

$\displaystyle \frac{7}{28}+\frac{20}{28}=\frac{27}{28}$

Now, in our original problem this is really is:

$\displaystyle \frac{\frac{27}{28}}{\frac{2}{5}}$

When you divide by a fraction, you really multiply by the reciprocal:

$\displaystyle \frac{\frac{27}{28}}{\frac{2}{5}}=\frac{27}{28} \cdot\frac{5}{2}=\frac{135}{56}$

Begin by simplifying the numerator and the denominator.

Numerator

$\displaystyle \frac{81}{5}+\frac{7}{25}$ has a common denominator of $\displaystyle 25$.  Therefore, we have:

$\displaystyle \frac{405}{25}+\frac{7}{25} = \frac{412}{25}$

Denominator

$\displaystyle \frac{1}{9}-\frac{2}{7}$ has a common denominator of $\displaystyle 63$.  Therefore, we have:

$\displaystyle \frac{7}{63}-\frac{18}{63}=-\frac{11}{63}$

Now, reconstructing our fraction, we have:

$\displaystyle \frac{\frac{412}{25}}{-\frac{11}{63}}$

To make this division work, you multiply the numerator by the reciprocal of the denominator:

$\displaystyle \frac{\frac{412}{25}}{-\frac{11}{63}} = \frac{412}{25} \cdot -\frac{63}{11} =-\frac{25956}{275}$

Recall that dividing is equivalent multiplying by the reciprocal.  Therefore, ((x - 4) / (1 / 2)) / (1 / (x + 4)) = ((x - 4) * 2)  *  (x + 4) / 1. 

Let's simplify this further:

(2x – 8) * (x + 4) = 2x² – 8x + 8x – 32 = 2x² – 32

Begin by isolating the variables:

$\displaystyle \frac{3x}{8}+2x=\frac{1}{3}-12$

Now, the common denominator of the variable terms is $\displaystyle 8$. The common denominator of the constant values is $\displaystyle 3$. Thus, you can rewrite your equation:

$\displaystyle \frac{3x}{8}+\frac{16x}{8}=\frac{1}{3}-\frac{36}{3}$

Simplify:

$\displaystyle \frac{19x}{8}=-\frac{35}{3}$

Cross-multiply:

$\displaystyle 19x * 3 = -35*8$

Simplify:

$\displaystyle 57x = -280$

Finally, solve for $\displaystyle x$:

$\displaystyle x = -\frac{280}{57}$

Wherever there is an x, plug in 11 and perform the given operations. The numerator will be equal to 83 and the denominator will be equal to 46. 83 divided by 46 is equal to 1.804… and since the second decimal place is not greater than or equal to 5, the first decimal place stays the same when rounding so the final answer is 1.8.

To solve for an inverse, we switch x and y and solve for y. Doing so yields:

$\displaystyle \frac{3x-20}{2}=y$

$\displaystyle 5x-4y=18$

1. Switch the $\displaystyle x$ and $\displaystyle y$ variables in the above equation.

$\displaystyle 5y-4x=18$

2. Solve for $\displaystyle y$:

$\displaystyle 5y-4x=18$

$\displaystyle 5y-4x+4x=18+4x$

$\displaystyle 5y=4x+18$

$\displaystyle \frac{5y}{5}=\frac{4x+18}{5}$

$\displaystyle y=\frac{4x+18}{5}$

If $\displaystyle y$ varies inversely with $\displaystyle x$, $\displaystyle y=\frac{K}{x}$.

1. Using any of the two $\displaystyle x,y$ combinations given, solve for $\displaystyle K$:

Using $\displaystyle (2,18)$:

$\displaystyle 18=\frac{K}{2}$

$\displaystyle K=36$

2. Use your new equation $\displaystyle y=\frac{36}{x}$ and solve when $\displaystyle x=12$:

$\displaystyle y=\frac{36}{12}=3$

An inverse variation is a function in the form: $\displaystyle xy = k$ or $\displaystyle y = \frac{k}{x}$, where $\displaystyle k$ is not equal to 0. 

Substitute each $\displaystyle \left ( x,y \right )$ in $\displaystyle xy = k$.

$\displaystyle 5(4.8) = 24$

$\displaystyle 6.4(3.75) = 24$

$\displaystyle 20(1.2) = 24$

Therefore, the constant of variation, $\displaystyle k$, must equal 24. If $\displaystyle y$ varies inversely as $\displaystyle x$, $\displaystyle 3n$ must equal 24. Solve for $\displaystyle n$.

$\displaystyle 3n = 24$

$\displaystyle n = 8$

If $\displaystyle x=3$ when $\displaystyle y=8$, and the variation is direct, then $\displaystyle xy = 24$. Using this, we know that if $\displaystyle y=6$, $\displaystyle x= \frac{24}{6} = 4$.