### Instruction: What is the largest value of $x$, if $\frac{x}{5} + \frac{1}{5x} = \frac{1}{2}$?### Response: To solve for $x$, I need to get rid of the fractions.
A common way to do that is to multiply both sides by the lowest common denominator of all the fractions.
In this case, that would be $10x$, since $10x$ is divisible by both $5$ and $5x$.
Multiplying both sides by $10x$ gives me $2x^2 + 2 = 5x$.
Rearranging the terms, I get a quadratic equation: $2x^2 - 5x + 2 = 0$.
To solve this, I can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = -5$, and $c = 2$.
Plugging in the values, I get $x = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm 3}{4}$.
This gives me two possible values for $x$: $\frac{5 + 3}{4} = 2$ and $\frac{5 - 3}{4} = \frac{1}{2}$.
The question asks for the largest value of $x$, so I choose $x = 2$.