Question

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

Answer 1

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.