sanju09 wrote:If x y z ≠ 0, is x (y + z) ≥ 0?

[1] |y + z| = |y| + |z|.

[2] |x + y| = |x| + |y|.



[spoiler]Source: www.avenuesabroad.org[/spoiler]

If any one of x, y or z is -ve, while both the others are =ve, then the answer is no.

If any one is negative, while both the others are +ve, while either has the same absolute value as the -ve no., then the answer is no.

If all are +ve, then the answer is yes.

If all are -ve, then the answer is no.

St1] Thie means that 'y' & 'z' share the same sign, while no info about 'x'. Insuff.

St2] This means that 'x' & 'y' share the same sign, while no info about z. Insuff

Together- all share same signs; however all could be +ve of -ve. Insuff.

I pick E

combining the two statements, x, y, z all have the same signs. x (y + z)>0 whether x,y,z are all positive or all negative

x (y + z)=> +(+)>0 or -(-)>0

hence, C

sanju09 wrote:If x y z ≠ 0, is x (y + z) ≥ 0?

[1] |y + z| = |y| + |z|.

[2] |x + y| = |x| + |y|.



[spoiler]Source: www.avenuesabroad.org[/spoiler]

Lets use Plug-In

(1)
For sufficing the inequality y and z has to be of the same sign.
x____y____z_____|y| + |z|___|y + z|___x (y + z)____x (y + z) ≥ 0
1____3____2_______5_______5_______5 __________YES
-1____3____2_______5_______5_______-5 __________NO
1___-3____-2_______5_______5 _______-5 __________NO
-1___-3____-2_______5_______5_______5 __________YES
INSUFFICIENT
It is only YES if x, y and z are of the same sign.


(2)
For sufficing the inequality x and y has to be of the same sign.
x____y____z____|x| + |y|_____|x + y|_____x (y + z)_____x (y + z) ≥ 0
2____3____4_______5_________5_________14 __________YES
2____3____-4______5_________5_________-2 __________ NO
-2___-3____4_______5_________5_________-2 __________NO
-2___-3____-4______5_________5_________14 __________YES
INSUFFICIENT
It is only YES if x, y and z are of the same sign.



(1) and (2) together:
For sufficing all the inequalities x, y and z has to be of the same sign.
x___y___z___|y| + |z|___|y + z|__|x| + |y|__|x + y|__x (y + z)__x (y + z) ≥ 0
2___3___2_____5________5______5_______5_____10_______YES
-2__-3___-2_____5_______5______5_______5_____10_______YES

SUFFICIENT
[spoiler]CORRECT ANSWER (C)[/spoiler]