Answer to Question #157291 in Calculus for Vishal

Question #157291

Show that for every real number x > 0, there exists a real number y > 0 such that |2x + y| = 5


1
Expert's answer
2021-01-25T14:36:03-0500

The formulation is incorrect. It should be as

"Show that for every real number x > 0, there exists a real number y < 0 such that |2x + y| = 5" or

"Show that for every real number x > 0, there exists a real number y > 0 such that |2x - y| = 5".

The solution is as follows.

|2x+y| = 5 implies "2x+y = \\pm5" and "y = \\pm5-2x".

If x>2.5 then both values of y are negative. So, the original statement is wrong.


Let's prove that for every real number x > 0, there exists a real number y > 0 such that |2x - y| = 5.

|2x-y| = 5 implies "2x-y = \\pm5" and "y = 2x\\mp 5".

If x>0 then y1=2x+5>0 and this corrected statement is proved.


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