Juan used the expression 16 – 9 – 12 + 22 to find his profit for days 2 and 3. He rewrote the expression as 16 + (–9) + (–12) + 22. Juan can use the associative and commutative properties to rewrite the expression again. Explain why he had to use the additive inverse before he could use these properties. Juan has $10 to spend. He needs $13 to purchase the supplies. Explain how to find the amount of money Juan is short.

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the industrial development of cities in the ruhr area in the nineteenth century is described. the research was based on english and german language sources. several factors that shaped growth and paths of the cities were categorized and analyzed. they are locations, resources, traditions, entrepreneurs, and technological developments. locations and resources were the most important among them; other factors may be regarded as consequences from locations and resources. among the cities, bochum, essen, gelsenkirchen, and dortmund had the richest resources and most favorable locations in the transportation network and grew to be the largest cities in the ruhr area near the end of the nineteenth century. riverside location was also a favorable factor for the development, as shown in the development of duisburg on the confluence of rhine and ruhr. resource distribution can explain the rise of textile industry, metallurgical industry, and coal and iron industries, and the northward progression of the mining industry. tradition was an important factor for cities such as krefeld, barmen, elberfeld, solingen, and remscheid, which experienced development based on their traditions through the nineteenth century. various entrepreneurs exerted influences within and beyond the local level over the ruhr area. technological development stimulated the northward progression of the mining industry and the deep connection between the coal and the iron industries.