He is VERY SKEPTICAL of this. "... Shouldn't you be hunting jewel shards instead?" Instead of getting killed.

"I'll do that after."

"Well, have fun ghost-hunting."

"Fine."

And she'll drop down into the well again and disappear.

Well, she shouldn't fine.

It's up to her, then, and apparently she's working on a deadline. She doesn't know how long it'll be until the Soul Piper opens its eyes, and in fact knows precious little about ghosts at all, but - she knows that they happen when someone is upset, when something about their life was left unresolved. You'd expect dead children to leave all sorts of loose ends when they died, having not accomplished any of the things they really wanted to, which is probably why the Soul Piper exists in the first place. 

Kagome... does not have the Soul Piper's expertise, here. But she knows where to find out more about anything that might specifically be bothering Mayu, and she can't help her let go of it until she knows what it is. She runs half the way from the train station to the hospital. She's getting better at running, she thinks. She does a lot of it, these days. Even so, it's past sunset by the time she gets there. 

She's barely begun speaking to Mayu's mother in the parking lot when one of the upper-story windows explodes, and something in her knows immediately what room it is.

Inuyasha is BORED. It has been, like, a whole thirty minutes, and he is SO BORED. BORED.

BOOOOOOOOOOOOOOOOOOOOOORED.

Well, through the portal with him!

And now for the hunt. Over the roofs of the town, swift as lightning, silent as a shadow, without a single person noticing (He thinks. What's a security camera?), after Kagome's scent.

Where is she when he catches up?