We begin by squaring both sides of the inequality. This does not change the inequality sign because we are told that √x is the positive root, so the sum of √x and 1/√x is positive, and squaring a positive number will not change the sense of the inequality. If √x + 1/√x < 4, then (√x + 1/√x)² < (4)².
So, we now have x + 2 + 1/x < 16
Next, we should rearrange the inequality by isolating either x or 1/x. I will isolate 1/x in my solution by adding - x - 2 to both sides to give - x - 2 + x + 2 + 1/x < 16 – x - 2, which simplifies to 1/x < 14 - x. The sense of the inequality remains unchanged either way. Proof:
Let Y = x + 2 + 1/x and Z = 16. We know now that Z < Y, so that Y - Z must be greater than 0.
(Y - x) - (Z - x) = Y - x - Z + x
-x and x cancel so we end up with Y - Z, which is greater than zero.
We should now multiply both sides by x to give 1 < 14x - x². Multiplication by a positive number also does not alter the sense of the inequality. Proof:
Let C = 1/x and D = 14x – x. We know that D > C, so that D - C > 0 when x is greater than 0.
When the all this is true, x (D - C) > 0, and Dx – Cx > 0, and so Dx > Cx.
1 < 14x – x² can be rearranged to give 0 < - x² + 14x – 1. We are now able to solve this inequality for x using the quadratic formula. In doing this, we obtain two x values, namely
7 + 4√3 and 7 - 4√3. At this point, I thought it was helpful to sketch a graph to help visualise the next step.
7 + 4√3 and 7 - 4√3 are the points at which the graph y = - x² + 14x - 1 crosses the line y = 0 (the x-axis). We want a set of x values which give the part of the graph that is ABOVE the x-axis. As this is a graph with a maximum turning point (negative x squared coefficient) which we know is above the x-axis as the graph crosses the x-axis, the full set of x values which satisfy our inequality must lie in between 7 + 4√3 and 7 - 4√3 (the two points which cross the x-axis).
Therefore, 7 - 4√3 < x < 7 + 4√3 is the range of x values for which √x + 1/√x < 4.