AP Calc
posted by Anon on .
Solve the following absolute value inequalities:
x+1<=x3
I promise I'm not posting my whole homework assignment. These are questions that I skipped and have no clue how to do out of a huge packet.

The key to absolute value problems is to remember the definition of n.
n = n if n >= 0
n = n if n < 0
So, here we have
x+1<=x3
if (x3) >= 0, x3 = x3
Also, since that means x > 3, x+1 >= 0, so x+1 = x+1 and we have
x+1 <= x3
1 < 3
not possible
If (x3) < 0, (that is, x<3) we have
x+1 <= (x3)
Now, if x >= 1, x+1 >= 0, and we have
x+1 <= x+3
2x <= 2
x <= 1
But, we required above that x >= 1, so we get a solution set 1 <= x <= 1
If (x3) < 0 and (x+1) < 0 we have
(x+1) <= (x3)
x1 <= x+3
1 < 3
So x < 1 is also a solution.
Combining the solution sets, we see that
x <= 1 is the complete solution set
To see how this works graphically, visit
http://rechneronline.de/functiongraphs/
and enter
abs(x+1)
abs(x3)
as your two functions, and set the domain for x from 5 to 5, and the range y from 0 to 5
This is an excellent place to try out viewing function graphs to confirm your algebra.