Thread: Solve THIS!

Here is a nice problem which gets many people. Not because it is tricky nor difficult.
Solve for
.
I think it gets people because they think you can solve it with basic algebraic moves.

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.

Originally Posted by Treadstone 71

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.

HA HA got you!! What about ?

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

You didn't specify to solve for ALL x, but to find a solution for the equation. I found two solutions.

'Solve for x' is generally used when you're asked to find all x which satisfy the equation or inequality.

Originally Posted by Treadstone 71

You didn't specify to solve for ALL x, but to find a solution for the equation. I found two solutions.

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.

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

Originally Posted by TD!

'Solve for x' is generally used when you're asked to find all x which satisfy the equation or inequality.

True, true