x=0, x=-1
You actually can solve it algebraically.
Assume x 0, |x|+|x+1|=2x+1=1 => 2x=0 => x=0.
Assume x<0, |x|+|x+1|=-x+(-x-1)=-2x-1=1=>-2x=2 => x=-1.
It can be done algebraically.
|x| + |x+1| = 1
(|x|+|x+1|)² = 1² (both sides are non-negative so squaring is allowed)
2|x(x + 1)| + 2x² + 2x + 1 = 1
2|x(x + 1)| = -2x²-2x
|x(x + 1)| = -x(x+1)
(|x(x + 1)|)² = (-x(x+1))² (*)
(x(x+1))² = (x(x+1))²
The last line is trivially true for all x, but there's an extra condition in (*).
You can only square if both sides have the same sign, the LHS is non-negative so the RHS has to be non-negative either. This gives the condition:
(x(x+1)) ≥ 0
With solution, after basic algebra: -1 ≤ x ≤ 0
Come on, Treadstone stop manipulating me! You knew well that you need to solve for all x. You just made a mistake and it happens. I also made many mistakes on these forums read through them and you will find them.Originally Posted by Treadstone 71
Anyways, I happen to like this equation because it solutions are not finite. Good job, TD!
The method I used is this:
If then .
Thus, thus, one solution.
If thus, then . Thus, thus, but thus no solutions.
If thus, but this is always true! Thus, are all solutions.
Now take the union of all the solution sets
Another, non-rigorous, way of solving this problem is with a graph. Graph and graph and see where the graphs intersect. You will see the graphs coincide on