{"id":1287,"date":"2023-04-04T19:41:25","date_gmt":"2023-04-04T19:41:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1287"},"modified":"2024-10-18T20:51:10","modified_gmt":"2024-10-18T20:51:10","slug":"algebraic-equations-learn-it-5","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/algebraic-equations-learn-it-5\/","title":{"raw":"Algebraic Equations: Learn It 5","rendered":"Algebraic Equations: Learn It 5"},"content":{"raw":"<p>Three cases can come up as we are solving linear equations - equations with one solution, equations with no solutions, and equations with infinite solutions. We have already seen where an equation has one solution. Let's explore equations that do not have any solutions and some that have an infinite number of solutions.<\/p>\r\n<h2>Equations with No Solutions<\/h2>\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<br \/>\r\n<center>[latex]12+2x\u20138=7x+5\u20135x[\/latex]<\/center>[reveal-answer q=\"790409\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"790409\"]Combine <b>like terms<\/b> on both sides of the equation.\r\n\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}12+2x-8=7x+5-5x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\end{array}[\/latex]<\/p>\r\n<p>Isolate the [latex]x[\/latex] term by subtracting [latex]2x[\/latex]<i><\/i> from both sides.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4= \\,5\\end{array}[\/latex]<\/p>\r\n<p>This false statement implies there are <strong>no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or [latex]DNE[\/latex] for short.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the example above, a solution was not obtained. Using the properties of equality to isolate the variable resulted instead in the false statement [latex]4=5[\/latex]. Certainly, [latex]4[\/latex] is not equal to [latex]5[\/latex].<\/p>\r\n<p>Note that in the second line of the solution above, the statement [latex]2x+4=2x+5[\/latex] was obtained after combining like terms on both sides. If we examine that statement carefully, we can see that it was false even before we attempted to solve it. It would not be possible for the quantity [latex]2x[\/latex] with [latex]4[\/latex] added to it to be equal to the same quantity [latex]2x[\/latex] with [latex]5[\/latex] added to it. The two sides of the equation do not balance. Since there is no value of [latex]x[\/latex] that will ever make this a true statement, we say that the equation has <i>no solution.<\/i><\/p>\r\n<p>Be careful that you do not confuse the solution [latex]x=0[\/latex] with <em>no solution.<\/em>\u00a0The solution [latex]x=0[\/latex]\u00a0means that the value [latex]0[\/latex] satisfies the equation, so there <i>is <\/i>a solution. To say that a statement has no solution means that there is no value of the variable, not even [latex]0[\/latex], which would satisfy the equation (that is, make the original statement true).<\/p>\r\n<p>As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance of a number from [latex]0[\/latex] on a number line, so the absolute value of a number must be a positive. When an absolute value expression is given to be equal to a negative number, we say the equation has no solution ([latex]DNE[\/latex], for short). Notice how this happens in the next example.<\/p>\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<br \/>\r\n<center>[latex]7+\\left|2x-5\\right|=4[\/latex]<\/center>[reveal-answer q=\"173733\"]Show Solution[\/reveal-answer][hidden-answer a=\"173733\"]Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.\r\n\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\r\n<p>Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or [latex]DNE[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Equations with Many Solutions<\/h2>\r\n<p>You have seen that if an equation has no solution, you end up with a false statement instead of a value for [latex]x[\/latex]. It is possible to have an equation where any value for [latex]x[\/latex] will provide a solution to the equation. <br \/>\r\n<br \/>\r\nIn the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.<\/p>\r\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<br \/>\r\n<center>[latex]5x+3\u20134x=3+x[\/latex]<\/center>[reveal-answer q=\"773733\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"773733\"]Combine like terms on both sides of the equation.\r\n\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}5x+3-4x=3+x\\\\x+3=3+x\\end{array}[\/latex]<\/p>\r\n<p>Isolate the [latex]x[\/latex] term by subtracting [latex]x[\/latex] from both sides.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3=3+x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,=\\,\\,3\\end{array}[\/latex]<\/p>\r\n<p>This true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as \"All Real Numbers.\"<br \/>\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>When solving, the true statement \u201c[latex]3=3[\/latex]\u201d was obtained. When solving an equation reveals a true statement like this, it means that the solution to the equation is <em>all real numbers<\/em>, that is, there are infinitely many solutions. Try substituting [latex]x=0[\/latex] into the original equation\u2014you will get a true statement! Try [latex]x=-\\dfrac{3}{4}[\/latex]. It will also satisfy the equation. In fact, any real value of [latex]x[\/latex] will make the original statement true.<\/p>","rendered":"<p>Three cases can come up as we are solving linear equations &#8211; equations with one solution, equations with no solutions, and equations with infinite solutions. We have already seen where an equation has one solution. Let&#8217;s explore equations that do not have any solutions and some that have an infinite number of solutions.<\/p>\n<h2>Equations with No Solutions<\/h2>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]12+2x\u20138=7x+5\u20135x[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q790409\">Show Solution<\/button><\/p>\n<div id=\"q790409\" class=\"hidden-answer\" style=\"display: none\">Combine <b>like terms<\/b> on both sides of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}12+2x-8=7x+5-5x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\end{array}[\/latex]<\/p>\n<p>Isolate the [latex]x[\/latex] term by subtracting [latex]2x[\/latex]<i><\/i> from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2x+4=2x+5\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-2x\\,\\,\\,\\,\\,\\,\\,\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4= \\,5\\end{array}[\/latex]<\/p>\n<p>This false statement implies there are <strong>no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or [latex]DNE[\/latex] for short.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In the example above, a solution was not obtained. Using the properties of equality to isolate the variable resulted instead in the false statement [latex]4=5[\/latex]. Certainly, [latex]4[\/latex] is not equal to [latex]5[\/latex].<\/p>\n<p>Note that in the second line of the solution above, the statement [latex]2x+4=2x+5[\/latex] was obtained after combining like terms on both sides. If we examine that statement carefully, we can see that it was false even before we attempted to solve it. It would not be possible for the quantity [latex]2x[\/latex] with [latex]4[\/latex] added to it to be equal to the same quantity [latex]2x[\/latex] with [latex]5[\/latex] added to it. The two sides of the equation do not balance. Since there is no value of [latex]x[\/latex] that will ever make this a true statement, we say that the equation has <i>no solution.<\/i><\/p>\n<p>Be careful that you do not confuse the solution [latex]x=0[\/latex] with <em>no solution.<\/em>\u00a0The solution [latex]x=0[\/latex]\u00a0means that the value [latex]0[\/latex] satisfies the equation, so there <i>is <\/i>a solution. To say that a statement has no solution means that there is no value of the variable, not even [latex]0[\/latex], which would satisfy the equation (that is, make the original statement true).<\/p>\n<p>As we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance of a number from [latex]0[\/latex] on a number line, so the absolute value of a number must be a positive. When an absolute value expression is given to be equal to a negative number, we say the equation has no solution ([latex]DNE[\/latex], for short). Notice how this happens in the next example.<\/p>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]7+\\left|2x-5\\right|=4[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173733\">Show Solution<\/button><\/p>\n<div id=\"q173733\" class=\"hidden-answer\" style=\"display: none\">Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\n<p>Result of absolute value is negative! The result of an absolute value must always be positive, so we say there is no solution to this equation, or [latex]DNE[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Equations with Many Solutions<\/h2>\n<p>You have seen that if an equation has no solution, you end up with a false statement instead of a value for [latex]x[\/latex]. It is possible to have an equation where any value for [latex]x[\/latex] will provide a solution to the equation. <\/p>\n<p>In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left leaves us with an equation with exactly the same terms on both sides of the equal sign.<\/p>\n<section class=\"textbox example\">Solve the following for [latex]x[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]5x+3\u20134x=3+x[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q773733\">Show Solution<\/button><\/p>\n<div id=\"q773733\" class=\"hidden-answer\" style=\"display: none\">Combine like terms on both sides of the equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}5x+3-4x=3+x\\\\x+3=3+x\\end{array}[\/latex]<\/p>\n<p>Isolate the [latex]x[\/latex] term by subtracting [latex]x[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3=3+x\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,=\\,\\,3\\end{array}[\/latex]<\/p>\n<p>This true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as &#8220;All Real Numbers.&#8221;\n<\/div>\n<\/div>\n<\/section>\n<p>When solving, the true statement \u201c[latex]3=3[\/latex]\u201d was obtained. When solving an equation reveals a true statement like this, it means that the solution to the equation is <em>all real numbers<\/em>, that is, there are infinitely many solutions. Try substituting [latex]x=0[\/latex] into the original equation\u2014you will get a true statement! Try [latex]x=-\\dfrac{3}{4}[\/latex]. It will also satisfy the equation. In fact, any real value of [latex]x[\/latex] will make the original statement true.<\/p>\n","protected":false},"author":15,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Beginning and Intermediate Algebra\",\"author\":\"Tyler Wallace\",\"organization\":\"\",\"url\":\"http:\/\/wallace.ccfaculty.org\/book\/book.html\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"learn_it","content_attributions":[{"type":"original","description":"Revision and Adaptation","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Beginning and Intermediate Algebra","author":"Tyler Wallace","organization":"","url":"http:\/\/wallace.ccfaculty.org\/book\/book.html","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program","author":"","organization":"Monterey Institute of Technology and Education","url":"http:\/\/nrocnetwork.org\/dm-opentext","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1287"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions"}],"predecessor-version":[{"id":15258,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1287\/revisions\/15258"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1287\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1287"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1287"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1287"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}