The figure shows 2 right triangles, triangle R S T with right angle S and triangle Q R T with right angle R. The measure of angle R T S is 60 degrees. The measure of angle Q T R is 45 degrees. The length of side R S is 2 square root 3. The length of side Q R is x. What is the value of x?

Consider right triangle RQT. In this triangle [tex] m\angle QRT=90^{\circ}, \ m\angle QTR=45^{circ} [/tex], then [tex] m\angle RQT=180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ} [/tex]. Thus the triangle RQT is isosceles and RQ=RT=x.
Consider right triangle RST. In this triangle [tex] m\angle RST=90^{\circ} , \ m\angle RTS=60^{\circ} [/tex], then [tex] m\angle TRS=180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ} [/tex]. The leg ST is opposite to the angle of 30°, so [tex] ST=\dfrac{RT}{2} =\dfrac{x}{2} [/tex]. 
For the right triangle RTS use the Pythagorean theorem:
[tex] RT^2=ST^2+RS^2,\\ x^2=(\sqrt{3})^2+\left(\dfrac{x}{2}\right)^2 ,\\ x^2=3+\dfrac{x^2}{4} ,\\ \dfrac{3x^2}{4} =3,\\ x^2=4,\\ x=2 [/tex].

Answer: x=2 units.