For 1., rearranging gives .

The LHS is a strictly increasing function in terms of x, so the minimal value occurs when x is minimized. If x = 0, the LHS = 1, but if x > 0, the LHS is greater than 1.

For 2., divide both sides by x:

.

Here, we want to show that the slope between two points (3, ln 3) and (x+3, ln (x+3)) is at most 1/3. This is true, because the derivative of y = ln x is 1/x, and the derivative is strictly decreasing. The maximal value of the LHS occurs when x gets infinitely close to 3, which is 1/3.