As Leonard and Sheldon are walking up the stairs to their apartment, they hear Penny singing horribly. When she sees them, she tells them that she's participating in a version of Rent and that they're both invited to come and see her. However, Leonard, not willing to go and hear her sing, lies that he and Sheldon have a conference to attend. Penny believes this, and Leonard thinks the matter is settled.
However, later on, Sheldon starts to get worried that Penny may do some research on her own, and find out that he and Leonard are not attending a conference after all. Therefore, he decides to fabricate an elaborate lie about he and Leonard going to Long Beach, to stage an intervention for Sheldon's-albeit-fake, cousin. Leonard gets upset at Sheldon for making such a lie, but Sheldon convinces him that it is more feasible. Penny believes this lie as well.
The cousin is Leopold, but answers to Leo or Lee. Sheldon believes Leonard would call him Lee. Sheldon also establishes a Facebook page which has "Lee's descent into drug addiction as well as a hopeful eHaromony listing. Thereby completing what Sheldon called an "un-unravel-able" lie.
Later, once again, Sheldon feels that the lie needs "touching up" and so asks a lab assistant to pose as his drug-addicted cousin, Leo. Sheldon does this, as he believes that if he "brings the cousin home," Penny will believe it even further.
In the end, it turns out that Penny recorded her performance and makes Leonard watch it, meaning that all of Sheldon's lies were of no use.