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
if (x-3) >= 0, |x-3| = x-3
Also, since that means x > 3, x+1 >= 0, so |x+1| = x+1 and we have
x+1 <= x-3
1 < -3
If (x-3) < 0, (that is, x<3) we have
|x+1| <= -(x-3)
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 (x-3) < 0 and (x+1) < 0 we have
-(x+1) <= -(x-3)
-x-1 <= -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
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.
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