{"id":642,"date":"2021-08-30T21:53:36","date_gmt":"2021-08-30T21:53:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=part&p=642"},"modified":"2021-09-19T23:01:22","modified_gmt":"2021-09-19T23:01:22","slug":"chapter-4-linear-equations-and-inequalities-in-one-variable","status":"publish","type":"part","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/part\/chapter-4-linear-equations-and-inequalities-in-one-variable\/","title":{"raw":"Chapter 4: Linear Equations and Inequalities in One Variable","rendered":"Chapter 4: Linear Equations and Inequalities in One Variable"},"content":{"raw":"

Why It Matters: Graphs<\/h2>\r\n
\r\n
\r\n

\""FallingWhy Graph Linear Relations and Inequalities?<\/h2>\r\nJust as the road sign at the right can help you quickly understand a \u201cfalling rocks\u201d hazard\u00a0ahead, graphs are a useful way of conveying information visually. Graphs of linear equations are especially effective for representing relationships between things that change at a constant rate, and they often do a better job than words or mathematical equations alone.\r\n\r\nRemember Joan? Joan was smart and got a ride home from the party. She picked up her car the next day and is happy that she made the choice not to drive intoxicated.\r\n\r\nThis week at work, Joan\u2019s manager has asked her\u00a0to create and present a financial report at the next company meeting. She wants to know how the amount of revenue Joan\u2019s team brought to the company changed during the last year. Her\u00a0job is to get the information\u00a0and choose the best method for communicating it.\r\n\r\nJoan starts with\u00a0a table\u00a0and adds the\u00a0values:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Month<\/th>\r\nRevenue<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/mn><\/mrow><\/semantics><\/math>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n1<\/mn>0<\/mn>0<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>0<\/span>0<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
1<\/mn><\/mrow><\/semantics><\/math>1<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n1<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
3<\/mn><\/mrow><\/semantics><\/math>3<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n1<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
5<\/mn><\/mrow><\/semantics><\/math>5<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n2<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>2<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
7<\/mn><\/mrow><\/semantics><\/math>7<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n2<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>2<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
9<\/mn><\/mrow><\/semantics><\/math>9<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n3<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>3<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
1<\/mn>1<\/mn><\/mrow><\/semantics><\/math>1<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n3<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>3<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table is nicely organized, and its\u00a0values are very precise, but it does not readily show\u00a0how the values\u00a0change over time. So she studies the numbers and tries to describe this change in\u00a0words:\r\n
\u201cOur team started the year with $<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span>$<\/mo><\/mrow>1000<\/mn><\/math><\/span><\/span> per month in revenue. For each additional month in the year\u00a02<\/mn>0<\/mn>1<\/mn>5<\/mn><\/mrow><\/semantics><\/math>2<\/span>0<\/span>1<\/span>5<\/span><\/span><\/span><\/span><\/span><\/span>, our team produced an increase in\u00a0$<\/span><\/span><\/span><\/span>250<\/span><\/span><\/span><\/span>$<\/mo><\/mrow>250<\/mn><\/math><\/span><\/span> of revenue.\u201d<\/span><\/div>\r\nNext, Joan tries expressing the information as a linear equation:\u00a0y<\/mi>=<\/mo>1<\/mn><\/mrow>4<\/mn><\/mrow><\/mfrac>x<\/mi>+<\/mo>1<\/mn>0<\/mn>0<\/mn>0<\/mn><\/mrow><\/semantics><\/math>y<\/span>=<\/span>\u200b<\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span>\u200b<\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span>\u200b<\/span><\/span><\/span><\/span>x<\/span>+<\/span>1<\/span>0<\/span>0<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nAgain, both are very accurate, but neither\u00a0reading two sentences out loud nor putting an equation on a poster board will go over very well with her manager or team members. She has not given them a way to picture the information. So, she tries graphing the equation which gives the following:\r\n
\"Graph\r\n

A graph showing the linear relationship between time and the change in revenue.<\/p>\r\n\r\n<\/div>\r\nAll four\u2014the table, the words, the equation, and the graph\u2014are representations of the same thing. And, depending on the purpose, one representation may get Joan\u2019s\u00a0point across more clearly than another.\r\n\r\nAs you can see, graphs are especially good\u00a0at conveying information about the relationship between things (in this case, time<\/em> and revenue<\/em>) with very little explanation. Joan\u2019s graph shows that the revenue her team produced during the last year steadily increased\u2014a very encouraging trend! Joan decides that the graph is definitely what she will present at the meeting.\r\n\r\nIn this module, you will learn how to create Cartesian graphs (like the one above) and why\u00a0graphs are an especially useful means of conveying information.\r\n\r\n<\/div>\r\n<\/div>","rendered":"

Why It Matters: Graphs<\/h2>\n
\n
\n

\""FallingWhy Graph Linear Relations and Inequalities?<\/h2>\n

Just as the road sign at the right can help you quickly understand a \u201cfalling rocks\u201d hazard\u00a0ahead, graphs are a useful way of conveying information visually. Graphs of linear equations are especially effective for representing relationships between things that change at a constant rate, and they often do a better job than words or mathematical equations alone.<\/p>\n

Remember Joan? Joan was smart and got a ride home from the party. She picked up her car the next day and is happy that she made the choice not to drive intoxicated.<\/p>\n

This week at work, Joan\u2019s manager has asked her\u00a0to create and present a financial report at the next company meeting. She wants to know how the amount of revenue Joan\u2019s team brought to the company changed during the last year. Her\u00a0job is to get the information\u00a0and choose the best method for communicating it.<\/p>\n

Joan starts with\u00a0a table\u00a0and adds the\u00a0values:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Month<\/th>\nRevenue<\/th>\n<\/tr>\n<\/thead>\n
0<\/mn><\/mrow><\/semantics><\/math>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n1<\/mn>0<\/mn>0<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>0<\/span>0<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
1<\/mn><\/mrow><\/semantics><\/math>1<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n1<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
3<\/mn><\/mrow><\/semantics><\/math>3<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n1<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>1<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
5<\/mn><\/mrow><\/semantics><\/math>5<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n2<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>2<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
7<\/mn><\/mrow><\/semantics><\/math>7<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n2<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>2<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
9<\/mn><\/mrow><\/semantics><\/math>9<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n3<\/mn>2<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>3<\/span>2<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n
1<\/mn>1<\/mn><\/mrow><\/semantics><\/math>1<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n3<\/mn>7<\/mn>5<\/mn>0<\/mn><\/mrow><\/semantics><\/math>3<\/span>7<\/span>5<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The table is nicely organized, and its\u00a0values are very precise, but it does not readily show\u00a0how the values\u00a0change over time. So she studies the numbers and tries to describe this change in\u00a0words:<\/p>\n

\u201cOur team started the year with $<\/span><\/span><\/span><\/span>1000<\/span><\/span><\/span><\/span>$<\/mo><\/mrow>1000<\/mn><\/math><\/span><\/span> per month in revenue. For each additional month in the year\u00a02<\/mn>0<\/mn>1<\/mn>5<\/mn><\/mrow><\/semantics><\/math>2<\/span>0<\/span>1<\/span>5<\/span><\/span><\/span><\/span><\/span><\/span>, our team produced an increase in\u00a0$<\/span><\/span><\/span><\/span>250<\/span><\/span><\/span><\/span>$<\/mo><\/mrow>250<\/mn><\/math><\/span><\/span> of revenue.\u201d<\/span><\/div>\n

Next, Joan tries expressing the information as a linear equation:\u00a0y<\/mi>=<\/mo>1<\/mn><\/mrow>4<\/mn><\/mrow><\/mfrac>x<\/mi>+<\/mo>1<\/mn>0<\/mn>0<\/mn>0<\/mn><\/mrow><\/semantics><\/math>y<\/span>=<\/span>\u200b<\/span>4<\/span><\/span><\/span><\/span>\u200b<\/span><\/span>\u200b<\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span>\u200b<\/span><\/span><\/span><\/span>x<\/span>+<\/span>1<\/span>0<\/span>0<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Again, both are very accurate, but neither\u00a0reading two sentences out loud nor putting an equation on a poster board will go over very well with her manager or team members. She has not given them a way to picture the information. So, she tries graphing the equation which gives the following:<\/p>\n

\"Graph<\/p>\n

A graph showing the linear relationship between time and the change in revenue.<\/p>\n<\/div>\n

All four\u2014the table, the words, the equation, and the graph\u2014are representations of the same thing. And, depending on the purpose, one representation may get Joan\u2019s\u00a0point across more clearly than another.<\/p>\n

As you can see, graphs are especially good\u00a0at conveying information about the relationship between things (in this case, time<\/em> and revenue<\/em>) with very little explanation. Joan\u2019s graph shows that the revenue her team produced during the last year steadily increased\u2014a very encouraging trend! Joan decides that the graph is definitely what she will present at the meeting.<\/p>\n

In this module, you will learn how to create Cartesian graphs (like the one above) and why\u00a0graphs are an especially useful means of conveying information.<\/p>\n<\/div>\n<\/div>\n","protected":false},"parent":0,"menu_order":18,"template":"","meta":{"pb_part_invisible":false,"pb_part_invisible_string":""},"contributor":[],"license":[],"class_list":["post-642","part","type-part","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/642","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/part"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/642\/revisions"}],"predecessor-version":[{"id":785,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/642\/revisions\/785"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=642"}],"wp:term":[{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=642"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=642"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}