$\displaystyle \frac{3\frac{1}{2}-2\frac{1}{5}}{4\frac{3}{4}}$

$\displaystyle =\left ( 3\frac{1}{2}-2\frac{1}{5} \right )\div 4\frac{3}{4}}$

$\displaystyle = \left ( \frac{7}{2}- \frac{11}{5} \right ) \div \frac{19}{4}$

$\displaystyle = \left ( \frac{35}{10}- \frac{22}{10} \right ) \times \frac{4}{19}$

$\displaystyle = \frac{13}{10} \times \frac{4}{19}$

$\displaystyle = \frac{13}{5} \times \frac{2}{19}$

$\displaystyle = \frac{26}{95}$

This is not among the given responses.

$\displaystyle \frac{3\frac{1}{2}+2\frac{1}{5}}{4\frac{3}{4}}$

$\displaystyle =\left ( 3\frac{1}{2}+2\frac{1}{5} \right )\div 4\frac{3}{4}}$

$\displaystyle = \left ( \frac{7}{2}+ \frac{11}{5} \right ) \div \frac{19}{4}$

$\displaystyle = \left ( \frac{35}{10}+ \frac{22}{10} \right ) \times \frac{4}{19}$

$\displaystyle = \frac{57}{10} \times \frac{4}{19}$

$\displaystyle = \frac{3}{5} \times \frac{2}{1}$

$\displaystyle = \frac{6}{5} = 1\frac{1}{5}$

Remember that that fraction bar is just a division symbol. Rewrite as a division, rewrite those mixed fractions as improper fractions, then rewrite as as a multiplication by the reciprocal of the second fraction.

$\displaystyle \frac{2\tfrac{1}{4}}{5\tfrac{2}{5}} = 2\tfrac{1}{4} \div 5\tfrac{2}{5} = \frac{9}{4} \div \frac{27}{5} = \frac{9}{4} \cdot \frac{5}{27} = \frac{1}{4} \cdot \frac{5}{3}= \frac{5}{12}$

Find the least common denominator which is $\displaystyle x^2$.

Just multiply the right fraction top and bottom by $\displaystyle x$. 

$\displaystyle \frac{2}{x^2}-\frac{4\cdot x}{x \cdot x}=\frac{2}{x^2}-\frac{4x}{x^2}$

Finally, subtract. 

Answer should be, 

 $\displaystyle \frac{2-4x}{x^2}$.

Lets try to factor. Remember, we need to find two terms that are factors of the c term that add up to the b term. 

$\displaystyle \frac{(x-2)(x-3)}{(x-2)(x+4)}-\frac{(x-3)(x-2)}{(x-3)(x+4)}$ 

Next cancel out the like terms.

$\displaystyle \frac{x-3}{x+4}-\frac{x-2}{x+4}$ 

Now combine the numerator. Remember to distribute the negative sign.

This is the final answer:

 $\displaystyle \frac{-1}{x+4}$.

Turn the $\displaystyle 1$ into a fraction that has a common denominator with the other fraction. To do this multiply $\displaystyle 1 \cdot \frac{x-3}{x-3}$.

This results in the following expression:

$\displaystyle \frac{x-3}{x-3}-\frac{3}{x-3}$

With the same denominator, just subtract and remember to distribute the negative sign.

The final answer is

 $\displaystyle \frac{x-6}{x-3}$.

Lets try to reduce the fraction. When factoring a difference of squares, the form is $\displaystyle (x-a)(x+a)$. When you foil, the middle terms cancel out.

So when doing that, we have:

$\displaystyle \frac{(x-3)(x+3)}{(x+3)}-\frac{(x-4)(x+4)}{(x+4)}$.

Cancel out like terms and we get this:

$\displaystyle (x-3)-(x-4)$.

Distribute the negative sign and we should get $\displaystyle 1$.

First find the least common denominator. That will be  $\displaystyle 2\sqrt{x}$ .

Multiply top and bottom of the left fraction by $\displaystyle 2$ .

$\displaystyle \frac{1}{\sqrt{x}}\cdot \frac{2}{2}=\frac{2}{2\sqrt{x}}$

Then subtract the numerator and then cross-multiply to get this: 

$\displaystyle \frac{2}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}=2\rightarrow 2-1=2(2\sqrt{x}) \rightarrow 1=4\sqrt{x}$

Then combine like terms. To get rid of the square root,

$\displaystyle \frac{1}{4}=\sqrt{x}$ square both sides to get $\displaystyle \left(\frac{1}{4}\right)^2=\frac{1}{16}$ as the final answer.

Find the least common denominator which is $\displaystyle x^2-4$. Then multiply the left fraction numerator by $\displaystyle x+2$ and multiply the right numerator by $\displaystyle x-2$ inorder for each fraction to share the common denominator.

$\displaystyle \frac{5(x+2)}{x^2-4}-\frac{2(x-2)}{x^2-4}=1$ 

Distribute and be careful of the negative sign.

$\displaystyle \frac{5x+10-2x+4}{x^2-4}=1$ 

Cross-multiply and create the quadratic equation.

$\displaystyle x^2-3x-18=0$ 

Lets factor. Remember, we need to find two terms that are factors of the c term that add up to the b term. 

$\displaystyle (x-6)(x+3)=0$

$\displaystyle x=6$, $\displaystyle x=-3$

If you check these answers back into the question, none of the fractions are undefined so these are the final answers. 

Find the least common denominator which is $\displaystyle x^2-1$. Then multiply the left fraction numerator by $\displaystyle x+1$ and multiply the right numerator by $\displaystyle x-1$ in order to make each fraction share the common denominator.

$\displaystyle \frac{2(x+1)}{x^2-1}-\frac{2(x-1)}{x^2-1}=\left | 2\right |$ 

Distribute and be careful of the negative sign.

$\displaystyle \frac{2x+2-2x+2}{x^2-1}=\left | 2\right |$ 

Simplify the numerator and cross multiply. Because there is an absolute value bar, we need to split this expression into two different equations.

Equation one:

$\displaystyle 2(x^2-1)=4$

$\displaystyle 2x^2-2=4$

$\displaystyle 2x^2=6$

$\displaystyle x^2=3$

$\displaystyle x=\pm \sqrt{3}$ By inspection, these values pass and don't violate the fractions being undefined.

Equation two:

$\displaystyle -2(x^2-1)=4$

$\displaystyle -2x^2+2=4$

$\displaystyle -2x^2=2$

$\displaystyle x^2=-1$

$\displaystyle x=i$ This answer is imaginary and not in the choices leaving $\displaystyle \pm \sqrt{3}$ as the answers.