document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172984>Question 1172984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A used furniture store buys old mattresses for $50 and sells them for $200. Write an inequality representing all the possible combinations of mattress purchases and sales that will result in a profit of less than $1500. Graph the inequality, and describe an aspect of the used furniture business that an inequality cannot account for.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850792 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850792\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define Variables**\r<BR>\n" );
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document.write( "* Let 'x' be the number of mattresses purchased.<BR>\n" );
document.write( "* Let 'y' be the number of mattresses sold.\r<BR>\n" );
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document.write( "**2. Calculate Profit per Mattress**\r<BR>\n" );
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document.write( "* Selling price: $200<BR>\n" );
document.write( "* Purchase price: $50<BR>\n" );
document.write( "* Profit per mattress: $200 - $50 = $150\r<BR>\n" );
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document.write( "**3. Write the Profit Inequality**\r<BR>\n" );
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document.write( "* Total profit from sales: 150y<BR>\n" );
document.write( "* Total cost of purchases: 50x<BR>\n" );
document.write( "* Net profit: 150y - 50x<BR>\n" );
document.write( "* We want the net profit to be less than $1500:<BR>\n" );
document.write( "    * 150y - 50x < 1500\r<BR>\n" );
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document.write( "**4. Simplify the Inequality**\r<BR>\n" );
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document.write( "* Divide the inequality by 50:<BR>\n" );
document.write( "    * 3y - x < 30\r<BR>\n" );
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document.write( "**5. Rewrite in Slope-Intercept Form (for graphing)**\r<BR>\n" );
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document.write( "* Add x to both sides:<BR>\n" );
document.write( "    * 3y < x + 30<BR>\n" );
document.write( "* Divide by 3:<BR>\n" );
document.write( "    * y < (1/3)x + 10\r<BR>\n" );
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document.write( "**6. Graph the Inequality**\r<BR>\n" );
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document.write( "* **Graph the boundary line:** y = (1/3)x + 10<BR>\n" );
document.write( "    * Y-intercept: (0, 10)<BR>\n" );
document.write( "    * Slope: 1/3 (rise 1, run 3)<BR>\n" );
document.write( "* **Dashed Line:** Since the inequality is strictly \"less than\" (<), use a dashed line to indicate that points on the line are *not* included in the solution.<BR>\n" );
document.write( "* **Shade the Region:** Shade the region *below* the dashed line, as this represents y < (1/3)x + 10.<BR>\n" );
document.write( "* **First Quadrant:** Remember that x and y must be non-negative (you can't buy or sell a negative number of mattresses), so the solution is restricted to the first quadrant.\r<BR>\n" );
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document.write( "**7. Aspect the Inequality Cannot Account For**\r<BR>\n" );
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document.write( "The inequality only accounts for the financial relationship between purchases, sales, and profit. It *cannot* account for:\r<BR>\n" );
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document.write( "* **Demand:** The inequality doesn't consider if there are enough customers willing to buy the mattresses. You might buy many mattresses but not be able to sell them all.<BR>\n" );
document.write( "* **Condition of Mattresses:** The inequality assumes all mattresses are sold for $200. Some mattresses might be in poor condition and need repairs, reducing profit, or they might not sell at all.<BR>\n" );
document.write( "* **Storage Costs:** The inequality does not factor in the cost of storing the mattresses.<BR>\n" );
document.write( "* **Time:** The inequality doesn't consider the time it takes to buy and sell the mattresses. If mattresses sit in storage for a long time, the opportunity cost increases.<BR>\n" );
document.write( "* **Market Fluctuations:** The inequality assumes a fixed selling price of $200. Market prices for used mattresses can change.<BR>\n" );
document.write( "* **Labor Costs:** The inequality does not factor in the labor costs associated with moving, cleaning, or repairing the mattresses.<BR>\n" );
document.write( "* **Disposal Costs:** Some mattresses might be too damaged to sell, and disposing of them could incur a cost.<BR>\n" );
document.write( "* **Inventory Limits:** The store may only have so much space for mattresses.<BR>\n" );
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