document.write( "Question 62974: A small company produces both doll houses and sets of doll furniture. The doll houses take 3 hours of labor to produce, and the furniture sets take 8 hours. The labor available is limited to 400 hours per week, and the total production capacity is 100 items per week. Existing orders requare that at least 20 doll houses and 10 sets of furniture be produced per week. Write a system of inequalities representing this situation, where x is the number of doll houses and y is the number of furniture sets. Then graph the system of inequalities. \n" ); document.write( "
Algebra.Com's Answer #43827 by Edwin McCravy(20054) $\"\"$ $\"About$
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A small company produces both doll houses and sets of doll furniture. The\r\n" );
document.write( "doll houses take 3 hours of labor to produce, and the furniture sets take 8\r\n" );
document.write( "hours. The labor available is limited to 400 hours per week, and the total\r\n" );
document.write( "production capacity is 100 items per week. Existing orders requare that at\r\n" );
document.write( "least 20 doll houses and 10 sets of furniture be produced per week. Write a\r\n" );
document.write( "system of inequalities representing this situation, where x is the number of\r\n" );
document.write( "doll houses and y is the number of furniture sets. Then graph the system of\r\n" );
document.write( "inequalities.\r\n" );
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document.write( "First we construct the labor hours inequality:\r\n" );
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document.write( "At 3 hours of labor for each of the x doll \r\n" );
document.write( "houses amounts to 3x hours of labor to make the \r\n" );
document.write( "x doll houses\r\n" );
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document.write( "At 8 hours of labor for each of the y furniture \r\n" );
document.write( "sets amounts to 8y hours of labor to make the y \r\n" );
document.write( "furniture sets.\r\n" );
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document.write( "So the total amount of labor is the sum of those, \r\n" );
document.write( "or 3x + 8y hours of labor\r\n" );
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document.write( "We are told that we are limited to 400 hours of \r\n" );
document.write( "labor so we write the labor hours inequality:\r\n" );
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document.write( "          3x + 8y < 400\r\n" );
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document.write( "Next we construct the items inequality:\r\n" );
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document.write( "The x doll houses amounts to x items per week\r\n" );
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document.write( "The y furniture sets amounts to y items per \r\n" );
document.write( "week.\r\n" );
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document.write( "So the total amount of items is the sum of \r\n" );
document.write( "those, or x + y items\r\n" );
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document.write( "We are told that we are limited to 100 items \r\n" );
document.write( "per week so we write the item inequality:\r\n" );
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document.write( "          x + y < 100\r\n" );
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document.write( "Next we construct the doll house inequality:\r\n" );
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document.write( "We are told that we must make at least 20 doll \r\n" );
document.write( "houses per week, so we write the doll house \r\n" );
document.write( "inequality:\r\n" );
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document.write( "          x > 20\r\n" );
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document.write( "Next we construct the furniture set inequality:\r\n" );
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document.write( "We are told that we must make at least 10 \r\n" );
document.write( "furniture sets per week, so we write the \r\n" );
document.write( "furniture set inequality:\r\n" );
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document.write( "          y > 10\r\n" );
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document.write( "Now there are two obvious inequalities which \r\n" );
document.write( "tell us that the number of doll houses and the \r\n" );
document.write( "number of furniture sets must not be a negative \r\n" );
document.write( "number, so they are these:\r\n" );
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document.write( "         x > 0\r\n" );
document.write( "         y > 0\r\n" );
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document.write( "So the six inequalities are\r\n" );
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document.write( "          3x + 8y < 400\r\n" );
document.write( "            x + y < 100\r\n" );
document.write( "                x > 20\r\n" );
document.write( "                y > 10\r\n" );
document.write( "                x > 0\r\n" );
document.write( "                y > 0\r\n" );
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document.write( "We draw the boundary lines, which are\r\n" );
document.write( "formed by replacing the inequality\r\n" );
document.write( "symbols by = signs, so we have these\r\n" );
document.write( "four boundary lines, and their intercepts:\r\n" );
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document.write( "                 3x + 8y = 400  (0,50),  (33.3,0)\r\n" );
document.write( "                   x + y = 100  (0,100), (100,0) \r\n" );
document.write( "                       x = 20   (20,0) vertical line\r\n" );
document.write( "                       y = 10   (0,10) horizontal line\r\n" );
document.write( "                       x = 0    y-axis \r\n" );
document.write( "                       y = 0    x-axis\r\n" );
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document.write( "Since x and y are non-negative we draw only the upper\r\n" );
document.write( "right quadrant, right of the y axis and above the x-axis:\r\n" );
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document.write( "Drawing the first boundary line in red:\r\n" );
document.write( "            \r\n" );
document.write( "Now since the origin (0,0) when substituted into the \r\n" );
document.write( "inequality 3x + 8y < 400 gives a true result\r\n" );
document.write( "0 < 400 the graph is underneath this line because \r\n" );
document.write( "that is the side of that line that the origin is on.\r\n" );
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document.write( "Now we draw the second boundary line in green:\r\n" );
document.write( "            \r\n" );
document.write( "Now since the origin (0,0) when substituted into the \r\n" );
document.write( "inequality x + y < 100 gives a true result\r\n" );
document.write( "0 < 100 the graph is underneath this line because \r\n" );
document.write( "that is the side of that line that the origin is on.\r\n" );
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document.write( "Now we draw the third boundary line in blue:\r\n" );
document.write( "            \r\n" );
document.write( "Now since the origin (0,0) when substituted into the \r\n" );
document.write( "inequality x < 20 gives a FALSE result\r\n" );
document.write( "0 > 20 the graph is to the right of this blue\r\n" );
document.write( "vertical line because that is the side the origin is\r\n" );
document.write( "NOT on.\r\n" );
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document.write( "Now we draw the fourth boundary line in purple:\r\n" );
document.write( "            \r\n" );
document.write( "Now since the origin (0,0) when substituted into the \r\n" );
document.write( "inequality y > 10 gives a FALSE result\r\n" );
document.write( "0 > 10 the graph is above this purple\r\n" );
document.write( "horizontal line because that is the side the \r\n" );
document.write( "origin is NOT on.\r\n" );
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document.write( "We do not have to graph the last two boundary lines,\r\n" );
document.write( "because they are the two axes, and we know that the\r\n" );
document.write( "region to shade is always above the x-axis and to the\r\n" );
document.write( "right of the y-axis.\r\n" );
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document.write( "So the area to be shaded is the region which is:\r\n" );
document.write( "1. below the red slanted line\r\n" );
document.write( "2. below the green slanted line\r\n" );
document.write( "3. right of the blue vertical line\r\n" );
document.write( "4. above the purple horizontal line\r\n" );
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document.write( "I can't shade that region on here, but you can on your \r\n" );
document.write( "paper. It's the 4-sided figure whose top side is red,\r\n" );
document.write( "bottom side purple, left side blue, and right side \r\n" );
document.write( "green.\r\n" );
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document.write( "Edwin
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