Traffic in optical backbone networks is increasing and becoming more heterogeneous with respect to bandwidth and QoS requirements due to the popularity of high-bandwidth services (such as cloud computing, e-science, telemedicine, etc.), which need to coexist with traditional services (HTTP, etc.). Mixed-line-rate (MLR) networks that support lightpaths of different rates such as 10, 40, 100 Gb/s, etc., are being studied to better support the heterogeneous traffic demands. Here, we study the important topic of degraded services in MLR networks, where a service can accept some degradation (i.e., reduction) in bandwidth in case of a failure in exchange for a lower cost, a concept called partial protection. Network operators may wish to support degraded services to optimize network resources and reduce cost. We propose using multipath routing to support degraded services in MLR networks, a problem that has not been studied before and is significantly more challenging than in single-line-rate (SLR) networks. We consider minimum-cost MLR network design (i.e., choosing which transponder rates to use at each node), considering the opportunity to exploit multipath routes to support degraded services. We propose a mixed-integer-linear-program (MILP) solution and a computationally efficient heuristic, and consider two partial-protection models. Our illustrative numerical results show that significant cost savings can be achieved due to partial protection versus full protection and is highly beneficial for network operators. We also note that multipath routing in MLR networks exploits volume discount of higher-line-rate transponders by cost-effectively grooming requests over appropriate line rates to maximize transponder reuse versus SLR.