intersting problem.

Posts: 791
Joined: Aug 4, 2011

by hemangsarkar, Sep 10, 2012, 6:42 AM

Q) if for all real $x$ ,
$|x + a - 3| + |x - 2a| = |2x - a - 3|$

find the set of values that $a$ can take.

my solution :
we start by noting that $x + a - 3 + x - 2a = 2x - a - 3$ .

so that implies $|m| + |n| = |m + n|$

this is true if $m$ and $n$ have the same sign. or if both of them or anyone of them is $0$ .

or, $mn \geq 0$

$x^2 - (3+a)x + 2a(3-a) \geq 0$

this should be true for all real values of $x$ ,

hence the discriminant must be $\le 0$ .

or, $(a+3)^2 - 8a(3-a) \le 0$

or, $(a-1)^2 \le 0$

so, $a = 1$