What about the boarding house rules makes it hard for Lyddie to have Rachel with her?

Children are not allowed at the worker’s boarding house and Rachel is too young to work.

Lyddie’s uncle shows up one day with her little sister and drops her off.  Lyddie was not expecting her, and has no idea what to do with her when she comes.  She has not seen her for two years.  The last she heard of Rachel was in her mother’s letter, where she indicated that Rachel was not doing well.

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Children are not allowed at the worker’s boarding house and Rachel is too young to work.

Lyddie’s uncle shows up one day with her little sister and drops her off.  Lyddie was not expecting her, and has no idea what to do with her when she comes.  She has not seen her for two years.  The last she heard of Rachel was in her mother’s letter, where she indicated that Rachel was not doing well.

When Lyddie introduces Rachel to Mrs. Bedlow, she doesn’t know what else to do.  She lives in the corporate boarding house.  She does not have anywhere else to put her little sister.

She could read the warning in Mrs. Bedlow's eyes. No men, no children (except for the keeper's own) in a corporation house. But surely the woman would not have the heart . . . (Ch. 19)

It is clear that Rachel is in bad shape.  She is very thin.  Lyddie asks Mrs. Bedlow if she can give her a bath because she is chilled.  She tells Mrs. Bedlow that she will get Rachel a job as a doffer, but Mrs. Bedlow tells her that Rachel is not old enough or strong enough.  Lyddie knows this.

"I'll pay, of course. Full board. And you see how small she is. You know she won't eat a full share." (Ch. 19)

Lyddie is afraid that she will lose her job because Rachel is staying with her.  However, her brother Charlie arrives to take her away.  He has found a family and the mother is interested in a daughter.  Lyddie is sad because she misses both of them, and Charlie taking Rachel seems to be the end of her chance for them to be a family again.

Even though Lyddie knows that Rachel is better off with Charlie and his new family, she is sad to see Rachel go.  All she wants is for Rachel to get stronger and go to school, but this is not going to happen if she lives with Lyddie.  All that she could possibly hope for is a factory job.