SOLUTION: Y is less than or equal to the absolute value of 4-X. I know what to do when X is not negative like if it was Y is less than or equal to the absolute value of X-4. Can you help?

Algebra ->  Absolute-value -> SOLUTION: Y is less than or equal to the absolute value of 4-X. I know what to do when X is not negative like if it was Y is less than or equal to the absolute value of X-4. Can you help?      Log On


   

Question 912598: Y is less than or equal to the absolute value of 4-X.
I know what to do when X is not negative like if it was Y is less than or equal to the absolute value of X-4. Can you help? Thank you.
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
y%3C=abs%284-x%29

Two cases. If 4-x%3E=0 and if 4-x%3C0.

CASE nonnegative argument
y%3C=abs%284-x%29
y%3C=4-x
y%3C=-x%2B4

CASE negative argument

y%3C=abs%284-x%29
y%3C=-4-%28-x%29
y%3C=-4%2Bx
y%3C=x-4

The solution is the region in the plane which ---deleted wording--- two cases.

SUMMARY OF THE INEQUALITES
Critical point seems to be for x=4.
highlight_green%28system%28y%3C=-x%2B4%2Cy%3C=x-4%29%29

The graphs show only the two individual cases; the ACTUAL solution is the region that ---deleted wording---.

Nonnegative argument case:
graph%28300%2C300%2C-8%2C8%2C-8%2C8%2Cy%3C=-x%2B4%29

Negative argument case:
graph%28300%2C300%2C-8%2C8%2C-8%2C8%2Cy%3C=x-4%29

What the solution region should look like:
graph%28300%2C300%2C-8%2C8%2C-8%2C8%2Cy%3C=abs%284-x%29%29

(About the deleted wording; either of the two individual inequalities will satisfy the absolute value inequality given. This is a union).