The least common denominator or the LCD of the given equation is the smallest number that can be a common denominator for the set of fractions. Usually, to get the LCD, we would multiply the denominators in the set of fractions.
Since the denominators are x, x + 10, and 3, we will multiply this and the LCD is 3x(x+10).
For the number of valid solutions, let's solve it and find how many,
[tex]\begin{gathered} \frac{1}{x}+\frac{2}{x+10}=\frac{1}{3} \\ \frac{x+10+2x}{x(x+10)}=\frac{1}{3} \\ \frac{3x+10}{x^2+10x}=\frac{1}{3} \\ 3(3x+10)=1(x^2+10x) \\ 9x+30=x^2+10x \\ 0=x^2+10x-9x-30 \\ x^2+x-30=0 \\ (x+6)(x-5)=0_{} \\ x=-6 \\ x=5 \end{gathered}[/tex]
Since there are two possible values of x, then there are 2 valid solutions.
