Can someone please explain to me how the answer is C? I dont understand how both inequalities together ensure that the answer to y comes out as 7

Thank you

There are 4 cases:

I x + 3 positive, 4 - x positive
II x + 3 positive, 4 - x negative
III x + 3 negative, 4 - x positive
IV x + 3 negative, 4 - x negative not possible because x could not be both <-3 and > 4


for I: y = 7
for II: y = 2x - 1
for III: y = 1 - 2x


with both (1) and (2) we know that case I applies


-- Veritas Help

This confuses me more than the answer in the back of the book... how do you come up with the equations for situations I, II, III? And how do you apply them to determine sufficiency? Don't we already know that situation I is true because of the absolute values?

if x + 3 is positive, |x+3| = x + 3
if 4 - x is positive, |4-x| = 4 - x

so |x+3| + |4-x| = x + 3 + 4 - x = 7


if x + 3 is positive, |x+3| = x + 3
if 4 - x is negative, |4-x| = -(4 - x) = x - 4

so |x+3| + |4-x| = x + 3 + x - 4 = 2x - 1

if x + 3 is negative, |x+3| = -(x + 3) = - x - 3
if 4 - x is positive, if 4 - x is positive, |4-x| = 4 - x

so |x+3| + |4-x| = -x - 3 + 4 - x = 1 - 2x

we need both statement (1) and statement (2) to know that x + 3 is positive and that 4 - x is positive, so case I applies.


Veritas Help