{"id":4415,"date":"2016-10-03T21:00:35","date_gmt":"2016-10-03T21:00:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&p=4415"},"modified":"2017-07-27T16:58:21","modified_gmt":"2017-07-27T16:58:21","slug":"conclusion-4","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/conclusion-4\/","title":{"raw":"Putting It Together: Graph Linear Equations and Inequalities","rendered":"Putting It Together: Graph Linear Equations and Inequalities"},"content":{"raw":"

Put it Together\r\n\"Watercolor<\/h2>\r\nJoan's uncle Devon is\u00a0a semiprofessional blogger who writes about media (he'd really like to be a paid\u00a0film critic, but no one has offered yet). Recently, Devon posted a short blog piece complaining\u00a0about the number of ads on TV these days, compared to\u00a0when he was younger. He wasn't very scientific about it, and a couple\u00a0of his\u00a0readers disputed a\u00a0claim he made and\u00a0tried to start an argument.\r\n\r\nDevon\u00a0asks\u00a0Joan, who is taking an algebra class at night, to\u00a0help him gather and cite some data to support his claim\u00a0that there are\u00a0\"way more\" TV ads now than when he was a kid. Together, they\u00a0get\u00a0online and do a little research, and find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009, shown below:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Years since 2009<\/th>\r\nMinutes<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n8.5<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n9.25<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n10<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n10.75<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n11.5<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n12.25<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n13<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe data show that, yes, there are more commercials now than in 2009. Joan decides the table isn't very exciting, because of the success she had with her manager when she reported her increase in sales using a graph.\u00a0Also, Devon\u00a0doubts that his readers will actually read a table. He agrees that a graph is\u00a0the perfect thing to post on his blog to convince his\u00a0skeptical readers!\r\n\r\nThey\u00a0get to work.\r\n\r\nFirst they \u00a0figure out what information they need to draw a graph of the line that represents the change in the number of minutes of commercials in one hour of TV since 2009?\r\n

The Cartesian Coordinate Plane<\/h3>\r\nJoan\u00a0remembers that the coordinate plane\u00a0gives graphs structure and meaning. A straight line on a page won't tell Devon's\u00a0readers much. Joan\u00a0draws the axes and labels the horizontal one \"Years Since 2009,\" because that's the first data point they have. \u00a0Then Devon labels the vertical axis from 1 to 18 because the minute data ranges from 8.5 to 13 minutes, and that will give room on either side.\r\n\r\n\"A\r\n\r\nThey plot the ordered pairs from the table of values on the\u00a0coordinate plane, as below.\r\n\r\n\"The\r\n\r\nThe points\u00a0give them a guide for drawing the line, which is shown below.\r\n\r\n\"An\r\n\r\nDevon is excited to\u00a0post the graph on his\u00a0blog to\u00a0show people how much more time they are being exposed to commercials in one hour of TV watching since 2009.\r\n\r\nThen, it happens . . .\r\n\r\nOne of his readers asks if he\u00a0can guess how many minutes of commercials will be in one hour of television ten years from now (assuming the current trend continues). After thinking about the question for a while, Joan\u00a0realizes they don't have to guess! \u00a0Joan and Devon have all the information they need to write the equation of the line they\u00a0drew. She tells Devon that he could then put in any value for the years since 2009.\r\n

Finding the Equation<\/h3>\r\nJoan remembers that knowing the slope and y-intercept of a line can help her write the equation of the line. \u00a0She realizes they know the y-intercept: [latex](0,8.5)[\/latex], and just need the slope.\r\n\r\nJoan checks her math notes to\u00a0find the definition of slope, and uses two data points to calculate it:\r\n\r\n[latex]\\begin{array}{l} \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}\\\\\\\\m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\\\\\m=\\frac{{11.5}-{10.75}}{{4}-{3}}\\\\\\\\m=\\frac{{0.75}}{{1}} = 0.75\\end{array}[\/latex]\r\n\r\nNow Devon and Joan have the two pieces of information they need to write the equation of the line that represents how many minutes of commercials will be in one hour of TV in any<\/em> year before or after 2009.\r\n\r\nFirst, they choose some variables: x<\/em> = the year and y<\/em> = the number of minutes. They then substitute the values for m<\/em> and b<\/em> into the slope-intercept form of a line:\r\n\r\n[latex]\\begin{array}{l}{ y }= {m x} + {b}\\\\{ y }= {0.75 x} + {8.5}\\end{array}[\/latex]\r\n\r\nRemembering that the whole point of this exercise was to answer Devon's\u00a0reader's question, they then\u00a0figure out what 10 years from now would be in relation to 0 representing 2009 on your graph. \u00a0If 10 years from now is 2026, then it's 17 years from 2009. \u00a0The x<\/em> value they need in order to answer the question is 17.\r\n\r\n[latex]\\begin{array}{l}{ y }= {0.75 (17)} + {8.5}\\\\\\\\{ y }= {12.75} + {8.5}\\\\\\\\{ y }= {21.25}\\end{array}[\/latex]\r\n\r\nDevon and Joan have a new data point: [latex](17,21.25)[\/latex]. \u00a0This means that in 2026 there will be more than\u00a020 minutes of commercials in one hour of TV. Yuck!\r\n\r\n \r\n\r\n ","rendered":"

Put it Together
\n\"Watercolor<\/h2>\n

Joan’s uncle Devon is\u00a0a semiprofessional blogger who writes about media (he’d really like to be a paid\u00a0film critic, but no one has offered yet). Recently, Devon posted a short blog piece complaining\u00a0about the number of ads on TV these days, compared to\u00a0when he was younger. He wasn’t very scientific about it, and a couple\u00a0of his\u00a0readers disputed a\u00a0claim he made and\u00a0tried to start an argument.<\/p>\n

Devon\u00a0asks\u00a0Joan, who is taking an algebra class at night, to\u00a0help him gather and cite some data to support his claim\u00a0that there are\u00a0“way more” TV ads now than when he was a kid. Together, they\u00a0get\u00a0online and do a little research, and find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009, shown below:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Years since 2009<\/th>\nMinutes<\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n8.5<\/td>\n<\/tr>\n
1<\/td>\n9.25<\/td>\n<\/tr>\n
2<\/td>\n10<\/td>\n<\/tr>\n
3<\/td>\n10.75<\/td>\n<\/tr>\n
4<\/td>\n11.5<\/td>\n<\/tr>\n
5<\/td>\n12.25<\/td>\n<\/tr>\n
6<\/td>\n13<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The data show that, yes, there are more commercials now than in 2009. Joan decides the table isn’t very exciting, because of the success she had with her manager when she reported her increase in sales using a graph.\u00a0Also, Devon\u00a0doubts that his readers will actually read a table. He agrees that a graph is\u00a0the perfect thing to post on his blog to convince his\u00a0skeptical readers!<\/p>\n

They\u00a0get to work.<\/p>\n

First they \u00a0figure out what information they need to draw a graph of the line that represents the change in the number of minutes of commercials in one hour of TV since 2009?<\/p>\n

The Cartesian Coordinate Plane<\/h3>\n

Joan\u00a0remembers that the coordinate plane\u00a0gives graphs structure and meaning. A straight line on a page won’t tell Devon’s\u00a0readers much. Joan\u00a0draws the axes and labels the horizontal one “Years Since 2009,” because that’s the first data point they have. \u00a0Then Devon labels the vertical axis from 1 to 18 because the minute data ranges from 8.5 to 13 minutes, and that will give room on either side.<\/p>\n

\"A<\/p>\n

They plot the ordered pairs from the table of values on the\u00a0coordinate plane, as below.<\/p>\n

\"The<\/p>\n

The points\u00a0give them a guide for drawing the line, which is shown below.<\/p>\n

\"An<\/p>\n

Devon is excited to\u00a0post the graph on his\u00a0blog to\u00a0show people how much more time they are being exposed to commercials in one hour of TV watching since 2009.<\/p>\n

Then, it happens . . .<\/p>\n

One of his readers asks if he\u00a0can guess how many minutes of commercials will be in one hour of television ten years from now (assuming the current trend continues). After thinking about the question for a while, Joan\u00a0realizes they don’t have to guess! \u00a0Joan and Devon have all the information they need to write the equation of the line they\u00a0drew. She tells Devon that he could then put in any value for the years since 2009.<\/p>\n

Finding the Equation<\/h3>\n

Joan remembers that knowing the slope and y-intercept of a line can help her write the equation of the line. \u00a0She realizes they know the y-intercept: [latex](0,8.5)[\/latex], and just need the slope.<\/p>\n

Joan checks her math notes to\u00a0find the definition of slope, and uses two data points to calculate it:<\/p>\n

[latex]\\begin{array}{l} \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}\\\\\\\\m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\\\\\m=\\frac{{11.5}-{10.75}}{{4}-{3}}\\\\\\\\m=\\frac{{0.75}}{{1}} = 0.75\\end{array}[\/latex]<\/p>\n

Now Devon and Joan have the two pieces of information they need to write the equation of the line that represents how many minutes of commercials will be in one hour of TV in any<\/em> year before or after 2009.<\/p>\n

First, they choose some variables: x<\/em> = the year and y<\/em> = the number of minutes. They then substitute the values for m<\/em> and b<\/em> into the slope-intercept form of a line:<\/p>\n

[latex]\\begin{array}{l}{ y }= {m x} + {b}\\\\{ y }= {0.75 x} + {8.5}\\end{array}[\/latex]<\/p>\n

Remembering that the whole point of this exercise was to answer Devon’s\u00a0reader’s question, they then\u00a0figure out what 10 years from now would be in relation to 0 representing 2009 on your graph. \u00a0If 10 years from now is 2026, then it’s 17 years from 2009. \u00a0The x<\/em> value they need in order to answer the question is 17.<\/p>\n

[latex]\\begin{array}{l}{ y }= {0.75 (17)} + {8.5}\\\\\\\\{ y }= {12.75} + {8.5}\\\\\\\\{ y }= {21.25}\\end{array}[\/latex]<\/p>\n

Devon and Joan have a new data point: [latex](17,21.25)[\/latex]. \u00a0This means that in 2026 there will be more than\u00a020 minutes of commercials in one hour of TV. Yuck!<\/p>\n

 <\/p>\n

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