To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{1}{2}=\frac{4}{x}\)

We have \(\displaystyle 2*4=1*x\). 

We get \(\displaystyle 8=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{1}{3}=\frac{5}{x}\)

We have \(\displaystyle 3*5=1*x\). 

We get \(\displaystyle 15=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{5}{4}=\frac{25}{2x}\)

We have \(\displaystyle 4*25=5*2x\).

We now have \(\displaystyle 100=10x\).

Divide both sides by \(\displaystyle 10\), we get \(\displaystyle 10=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{9}{4}=\frac{18}{2x}\)

We have \(\displaystyle 4*18=9*2x\).

We now have \(\displaystyle 72=18x\).

Divide both sides by \(\displaystyle 18\), we get \(\displaystyle 4=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{3}{7}=\frac{x}{2}\)

We have \(\displaystyle 3*2=7*x\).

We now have \(\displaystyle 6=7x\).

Divide both sides by \(\displaystyle 7\), we get \(\displaystyle \frac{6}{7}=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{x}{9}=\frac{4}{13}\)

We have \(\displaystyle 4*9=13*x\).

We now have \(\displaystyle 36=13x\).

Divide both sides by \(\displaystyle 13\), we get \(\displaystyle \frac{36}{13}=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{6}{x}=\frac{x}{6}\)

We have \(\displaystyle 6*6=x*x\).

We now have \(\displaystyle 36=x^2\). 

Take square root on both sides. Remember, when we do that the answer is \(\displaystyle \pm \sqrt{36}\).

Two negative or two positive numbers multiplied can give us a positve value.  \(\displaystyle 6\) is the answer to that perfect square and we add the positive and negative sign to get \(\displaystyle \pm6\).

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{x}{-3}=\frac{27}{-x}\)

We have \(\displaystyle -3*27=-x*x\).

When multiplying a negative and positive number, our answer is negative.

We now have \(\displaystyle -81=-x^2\).

We can divide both sides by \(\displaystyle -1\) to get \(\displaystyle 81=x^2\). 

Take square root on both sides. Remember, when we do that the answer is \(\displaystyle \pm \sqrt{81}\). Two negative or two positive numbers multiplied can give us a positve value.  \(\displaystyle 9\) is the answer to that perfect square and we add the positive and negative sign to get \(\displaystyle \pm9\).

To solve for \(\displaystyle x\), we just cross-multiply.

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

We have \(\displaystyle 2*5=1*(x+4)\).

Remember we treat the expression and place into a parantheses. We take care of the parantheses first and distribute the \(\displaystyle 1\).

We have \(\displaystyle 10=x+4\).

Subtract both sides by \(\displaystyle 4\). 

We get \(\displaystyle 6=x\). 

To solve for \(\displaystyle x\), we just cross-multiply.

\(\displaystyle \frac{6}{7}=\frac{12}{x-2}\)

We have \(\displaystyle 7*12=6*(x-2)\).

Remember we treat the expression and place into a parantheses. We take care of the parantheses first and distribute the \(\displaystyle 6\).

We have \(\displaystyle 84=6x-12\).

Add both sides by \(\displaystyle 12\).

We have \(\displaystyle 6x=96\). 

Then divide both sides by \(\displaystyle 6\). 

We get \(\displaystyle 16=x\).