Let's analyze the equation step by step:
Given the equation:
[tex]\[
\frac{1}{x} + \frac{2}{a+10} = \frac{1}{3}
\][/tex]
1. Finding the Least Common Denominator (LCD):
- To combine and compare fractions, we need to find the least common denominator (LCD) of the fractions involved in the equation.
- The denominators in the equation are [tex]\(x\)[/tex], [tex]\(a + 10\)[/tex], and 3.
- Therefore, the least common denominator (LCD) among these three denominators is 1 because the individual denominators are distinct and cannot simplify further to a common factor.
2. Determining the Number of Valid Solutions:
- The equation given is a rational equation.
- Generally, rational equations like the one given can have 2 solutions, which may include complex solutions.
- Hence, the number of valid solutions for this equation is 2.
So, the completed statements are:
- The least common denominator between all three fractions is [tex]\( \boxed{1} \)[/tex].
- The equation will have [tex]\( \boxed{2} \)[/tex] valid solutions.
