powerprep inequality problem: if x and y are integers

[i.ahmed111]

if x and y are integers and y=|x+3| +|4-x|, does y equal 7?

1) x <4
2) x>-3

I understand why the answer is c....any numbers used for x between the combined inequality range generates and answer of 7 for y. I got to the answer my picking numbers, but recognizing the trend took a little bit of time. Is there an elegant way to quickly attack inequality problems like these, or is a more conceptual number picking approach preferred, since you are dealing with two variables and more than 1 inequality?

Thanks!

[RonPurewal]

Is there an elegant way to quickly attack inequality problems like these, or is a more conceptual number picking approach preferred, since you are dealing with two variables and more than 1 inequality?

Thanks!

personally, i find number picking to be MORE "elegant", but that's not the issue.

you're not really dealing with two variables here. y isn't really a "variable" of its own; it's just an annoying name for the quantity |x + 3| + |4 - x|. so really, there's only one variable in this problem.

i.e., the problem is really just
If x is an integer, then is |x + 3| + |4 - x| = 7 ?
(we don't even need the stipulation that y is an integer, since that's guaranteed if x is an integer)

--

the algebraic method requires that you recognize where these quantities change signs.

BACKGROUND FACT:
if "QUANTITY" is POSITIVE OR ZERO, then |QUANTITY| = QUANTITY. i.e., strip the absolute-value bars, but leave the quantity alone.
if "QUANTITY" is NEGATIVE, then |QUANTITY| = -(QUANTITY). i.e., strip the bars and flip ALL the signs.

therefore:
consider |x + 3|
if (x + 3) is POSITIVE OR ZERO, then this will just be x + 3.
if (x + 3) is NEGATIVE, then this will be -x - 3.
so this is:
x + 3, for all x that are -3 or greater
-x - 3, for all x that are less than -3

consider |4 - x|
if (4 - x) is POSITIVE OR ZERO, then this will just be 4 - x.
if (4 - x) is NEGATIVE, then this will be -4 + x.
so this is:
4 - x, for all x that are 4 or less
-4 + x, for all x that are greater than 4

so if you have the two statements together, then it's just (x + 3) + (4 - x), or 7.

--

of course, it's MUCH easier just to plug in numbers, as you did, on this one.
not even close.

[i.ahmed111]

makes sense, thanks Ron!

[Ben Ku]

Glad it helped.

[vibhusethi]

makes sense

[mschwrtz]

We're glad that Ron could help.