The answer to this really lies in the answer to the question, “Where do you work”? It’s not good at all to live in South Bay if you have a job in Pasadena. It’s stupid, in fact. The one thing that grinds people down more than anything here is the traffic. It’s the worst thing about living here. So rather than worrying about “cheap and safe”, your FIRST order of business should be getting a place close to your job. THEN you can worry about safety and price. But, to reiterate, nothing is cheap here.
"LA" is such a big place, there are so many neighborhoods/cities where you can live. Of course, even within a city or neighborhood, there are safer sections and less-safe sections.
In Los Angeles, some nice sections are West LA, Brentwood, Westwood, Marina Del Rey, Playa Del Rey, Los Feliz, Silverlake, and Eagle Rock. Palms and Mar Vista are pretty good, too. In the Valley(part of LA), you have Encino, Tarzana, Studio City, Toluca Lake, Granada Hills, Woodland Hills, Sherman Oaks, West Hills, and Chatsworth. Glendale and Burbank are good places, and are incorporated cities of their own.
To the east: South Pasadena, parts of Pasadena, Altadena, Arcadia, Monrovia, Glendora, San Dimas, Laverne, Azusa, Rancho Cucamonga.
Along the beach: Santa Monica, Marina Del Rey, Playa Del Rey, Redondo Beach, Manhattan Beach, Hermosa Beach, Torrance, Palos Verdes, Rancho PV.
In Orange County aka "The OC": Seal Beach , Huntington Beach , Newport Beach , Corona Del Mar , Laguna Beach , Dana Point , Capistrano Beach , San Clemente , Brea, Yorba Linda, Orange, Tustin, Irvine, Laguna Hills, Rancho Santa Margarita, Lake Forest
To the west: Agoura, Calabasas, Thousand Oaks, Westlake Village, Simi Valley, Moorpark.
This is not an all-inclusive list, but it's a start. If you want to see the safety and price factors, go to www.lalife.com. Also, the LA Times did a nice job of outlining all 277 neighborhoods in LA County: http://projects.latimes.com/mapping-la/neighborhoods/.
Time to get researching!