| x + 1 | <= | x - 3 |
With two absolute values, there are 4 possible solutions:
First: x + 1 <= x - 3
Second: x + 1 <= -(x - 3)
Third: -(x + 1) <= x - 3
Fourth: -(x + 1) <= -(x - 3)
Let's work them out one at a time:
.
First: x + 1 <= x - 3
subtract x from both sides
1 <= -3
Never true. This one can never happen.
.
Second: x + 1 <= -(x - 3)
x +1 <= -x + 3
add x to both sides
2x + 1 <= 3
subtract 1 from both sides
2x < = 2
Answer: x < = 1
.
Third: -(x + 1) <= x - 3
-x - 1 <= x - 3
add x to both sides
-1 <= 2x - 3
add 3 to both sides
2 <= 2x
divide by 2 on both sides
1 <= x
Answer: x >= 1
.
Fourth: -(x + 1) <= -(x - 3)
-x - 1 <= -x + 3
add x to both sides
-1 <= 3
Never true. This solution can't happen.
.
So our 4 solutions are:
Can't happen, x <= 1, x >= 1, can't happen.
Either of the two solutions (the ones that can happen) can be true and the original equation is true.
But our two solutions (x <= 1, x >=1) cover everything- there is no value for x that doesn't fit into one of those solutions. That means x can be anything.
Answer: x can be any number.