Hello Archana 

X represent the strength of a solution/chemical. 10X means the solution is ten times higher in the solute concentration.

Example:

If i have a solution of sugar with initial concentration of 1mg/ml and someone ask me to prepare a 10X solution. I simply add 10 times sugar  (solute)  to the existing amount of solvent. The resultant solution become 10X.

Same as if someone ask me to make 1X solution from a 10X solution. I would dilute the existing solution 10 times by adding the water (solvent) to get the desired strength.

Normally in laboratory people prepare higher concentration of stock solutions (of antibiotics in your case) of about 10X, and during experimentation dilute it to working concentration 1X to avoid microbial contamination.  

Suppose you require a working concentration of 1mg/ml of the antibiotic solution, regard it as a 1X. Add ten time the solute (antibiotic), to existing amount of solvent, you will get solution of resultant strength of 10mg/ml, this would be your 10X antibiotic solution.

I think you would understand the concept. If you need further clarification, you can reply.

All the Best !!         

Thank you !!! Shashi Bhushan

Good explaination

Very good explanaition!