No. The Coachella Valley is definitely NOT in the Inland Empire, hence why locals, including myself, sometimes use the term "Desert Empire" to make it its own region. Once you drive past the town of Whitewater (next to Palm Springs) on the I-10, you will enter the Inland Empire region by going up a "hill" and entering through the town of Cabazon. The Inland Empire is generally composed of cities of western Riverside County and southwestern San Bernardino County. The westernmost IE city is considered to be Montclair (in SB County, next to Upland). Once past the Los Angeles County line or Orange County line, you have entered the LAMSA (Los Angeles Metropolitan Statistical Area), which composes the entirety of both LA County and Orange County. In conclusion, the Coachella Valley (Desert Empire) is its own region, NOT in the Inland Empire. Hope I helped! ;)
