SOLUTION: Show that 1/x <= 3 - 2sqrt(x) for all positive real numbers x. Describe when we have equality.

Question 1178757: Show that 1/x <= 3 - 2sqrt(x) for all positive real numbers x. Describe when we have equality.
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

Since x is positive we can multiply through by x without
changing the inequality sign.













1 | 4  -9   6  -1
  |     4  -5   1
    4  -5   1   0







The critical numbers are 1 and 1/4

The solution is (0,1/4) U {1}

However the interval (0,1/4) is ruled out by substituting
the test value x = 0.1 from that interval in the original
inequality:









which is clearly false.

Thus the only solution is {1} and that is when equality holds.

In other words the inequality



is equivalent to the equation:



because the left side can NEVER be less than the right side.

So we can ONLY and ALWAYS have equality, and x=1 is the ONLY
value x can take on

Edwin

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