What is the minimum value of $|2x-5|+6$?

Question

What is the minimum value of $|2x-5|+6$?

Answer 1

Hint. The absolute value is non-negative, hence $|2x-5|\geq 0$.

Answer 2

With x set as 2.5, the expression would equal 6

Absolute value only returns only floats equal or greater than 0. That being said, the range of this function is:

(0)+6 to (infinity)+6

with the latter being greater obviously because: 6 < Infinity+6

The minimum value would have to make the inside of the function equal 0 which means: $ |2x-5| = 0 $

And since it is zero, we can remove the absolute value sign to get:

$ 2x-5 = 0 $

$ 2x = 5 $

$ x = 2.5 $

Resulting in the expression to be:

$ |2x-5|+6 $

$ |2(2.5)-5|+6 $

$ |5-5|+6 $

$ |0|+6 $

$ = 6 $

Proof: http://i.imgur.com/CB5pwi6.png

EDIT: Accidentally typed former instead of latter.

EDIT: Added Image Proof.

EDIT: Used Expression signs to format instead of quotes.

Answer 3

We have $|2x+5|\geq 0$ for all values of x.

Therefore, $|2x+5|+6 \geq 6$ for all values of $x$.

$|2(-2.5)+5|+6=6$, so this minimum value is achieved at $x=-2.5$.

The minimum of $|2x+5|+6$ is thus $\boxed{6}$.