### Instruction: What is the largest value of $x$, if $\frac{x}{5} + \frac{1}{5x} = \frac{1}{2}$?### Response: To solve for $x$, I want to clear the fractions by multiplying both sides of the equation by the least common multiple of the denominators.
The least common multiple of 5, 5x, and 2 is 10x, so I multiply both sides by 10x and get: $2x^2 + 2 = 5x$.
I want to make this a quadratic equation in standard form, so I subtract 5x from both sides and get: $2x^2 - 5x + 2 = 0$.
To solve for $x$, I can either factor this quadratic or use the quadratic formula.
I notice that this quadratic can be factored as $(2x - 1)(x - 2) = 0$, which makes it easier to find the values of $x$ that make it zero.
By the zero product property, I can set each factor equal to zero and get: $2x - 1 = 0$ or $x - 2 = 0$.
Solving for $x$, I get: $x = \frac{1}{2}$ or $x = 2$.
However, the question asks for the largest value of $x$, so I only need to consider the value of $x$ that is greater than the other one.
Therefore, the largest value of $x$ is 2.