{"id":79,"date":"2023-06-21T13:22:30","date_gmt":"2023-06-21T13:22:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/review-topic-3\/"},"modified":"2024-01-08T18:47:18","modified_gmt":"2024-01-08T18:47:18","slug":"review-topic-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/review-topic-3\/","title":{"raw":"R1.3   Classifying Solutions to Linear Equations","rendered":"R1.3   Classifying Solutions to Linear Equations"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve equations that have one solution, no solution, or an infinite number of solutions<\/li>\r\n \t<li>Recognize when a linear equation that contains absolute value does not have a solution<\/li>\r\n \t<li>Classify an equation as a conditional equation, an inconsistent equation, or an identity.<\/li>\r\n<\/ul>\r\n<\/div>\r\nThere are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions.\r\n<h2>Equations with No Solutions<\/h2>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\r\n<p style=\"padding-left: 30px;\">[latex]12+2x\u20138=7x+5\u20135x[\/latex]<\/p>\r\n[reveal-answer q=\"790409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"790409\"]\r\n\r\nCombine <b>like terms<\/b> on both sides of the equation.\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\nIsolate the <i>x<\/i> term by subtracting 2<i>x<\/i> from both sides.\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\nThis false statement implies there are <strong>no solutions<\/strong> to this equation. Sometimes, we say the solution does not exist, or DNE for short.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example above, a solution was not obtained. Using the the properties of equality to isolate the variable resulted instead in the false\u00a0statement [latex]4=5[\/latex]. Certainly, [latex]4[\/latex] is not equal to [latex]5[\/latex].\r\n\r\nNote 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.\u00a0Since there is no value of <i>x <\/i>that will ever make this a true statement, we say that the equation has\u00a0<i>no solution.\u00a0<\/i>\r\n\r\nBe 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).\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Think\u00a0About It<\/h3>\r\nTry solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?\r\n\r\na) Solve [latex]8y=3(y+4)+y[\/latex]\r\n\r\nUse the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"933839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933839\"]\r\n<p style=\"text-align: center;\">Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\r\nFirst, distribute the 3 into the parentheses on the right-hand side.\r\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\r\nNext, begin combining like terms.\r\n<p style=\"text-align: center;\">[latex]8y=4y+12[\/latex]<\/p>\r\nNow move the variable terms to one side. Moving the [latex]4y[\/latex] will help avoid a negative sign.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,8y=4y+12\\\\\\underline{-4y\\,\\,-4y}\\\\\\,\\,\\,\\,4y=12\\end{array}[\/latex]<\/p>\r\nNow, divide each side by [latex]4y[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{4y}{4}\\normalsize =\\Large\\frac{12}{4}\\normalsize\\\\y=3\\end{array}[\/latex]<\/p>\r\nBecause we were able to isolate <em>y<\/em> on one side and a number on the other side, we have one solution to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\nb) Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]\r\n\r\nUse the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"937839\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937839\"]\r\n\r\nSolve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex].\r\n\r\nFirst, distribute the 2 into the parentheses on the left-hand side.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\r\nNow begin simplifying. You can combine the <em>x<\/em> terms on the left-hand side.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-10=2x+7\\end{array}[\/latex]<\/p>\r\nNow, take a moment to ponder this equation. It\u00a0says that [latex]2x-10[\/latex] is equal to [latex]2x+7[\/latex]. Can some number times two minus 10 be equal to that same number times two plus seven?\r\n\r\nPretend [latex]x=3[\/latex].\r\n\r\nIs it true that\u00a0[latex]2\\left(3\\right)-10=-4[\/latex] is equal to\u00a0[latex]2\\left(3\\right)+7=13[\/latex]. NO! We do not even really need to continue solving the equation, but we can just to be thorough.\r\n\r\nAdd [latex]10[\/latex] to both sides.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-10=2x+7\\,\\,\\\\\\,\\,\\underline{+10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\2x=2x+17\\end{array}[\/latex]<\/p>\r\nNow subtract [latex]2x[\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,2x=2x+17\\\\\\,\\,\\underline{-2x\\,\\,-2x}\\\\\\,\\,\\,\\,\\,\\,\\,0=17\\end{array}[\/latex]<\/p>\r\nWe know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that <em>x<\/em> could be to make this equation true.\r\n\r\nThis false statement implies there are <strong>no solutions<\/strong> to this equation, or DNE (does not exist) for short.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Equations with Many Solutions<\/h2>\r\nYou have seen that if an equation has no solution, you end up with a false statement instead of a value for <i>x<\/i>. It is possible to have an equation where any value for <em>x<\/em> will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left\u00a0leaves us with an equation with exactly the same terms on both sides of the equal sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\r\n<p style=\"padding-left: 30px;\">[latex]5x+3\u20134x=3+x[\/latex]<\/p>\r\n[reveal-answer q=\"773733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"773733\"]Combine like terms on both sides of the equation.\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\nIsolate the <i>x<\/i> term by subtracting <i>x<\/i> from both sides.\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\nThis true statement implies there are an infinite number of solutions to this equation, or we can also write the solution as \"All Real Numbers\"\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen 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]\u00a0into 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 x will make the original statement true.\r\n\r\nIndeed, after combining like terms, the equation [latex]x+3=3+x[\/latex] was obtained. It is certainly true that the quantity [latex]x[\/latex] with [latex]3[\/latex] added to it is equal to [latex]3[\/latex] with [latex]x[\/latex] added to it by the commutative property of addition.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\r\n<p style=\"padding-left: 30px;\">[latex]3\\left(2x-5\\right)=6x-15[\/latex]<\/p>\r\n[reveal-answer q=\"973733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"973733\"]\r\n\r\nDistribute the [latex]3[\/latex] through the parentheses on the left-hand side.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{r}3\\left(2x-5\\right)=6x-15\\\\6x-15=6x-15\\end{array}[\/latex]<\/p>\r\nWait! This looks just like the previous example. You have the same expression on both sides of an equal sign. \u00a0No matter what number you choose for <em>x<\/em>, you will have a true statement. We can finish the algebra:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6x-15=6x-15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-6x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-15\\,\\,=\\,\\,-15\\end{array}[\/latex]<\/p>\r\nThis true statement implies there are an infinite number of solutions to this equation.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for demonstrations of equations with <em>no solutions<\/em> and\u00a0<em>infinitely many solutions<\/em>.\r\n\r\nhttps:\/\/youtu.be\/iLkZ3o4wVxU\r\n\r\nThe next video demonstrates equations with no or infinitely many solutions involving parentheses.\r\n\r\nhttps:\/\/youtu.be\/EU_NEo1QBJ0\r\n<h2><span style=\"color: #ff9900;\">[Optional]<\/span> Absolute Value Equations with No Solutions<\/h2>\r\nAs we are solving absolute value equations, it is important to be aware of special cases. An absolute value is defined as the distance\u00a0 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 (DNE, for short). Notice how this happens in the next two examples.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\r\n<p style=\"padding-left: 30px;\">[latex]7+\\left|2x-5\\right|=4[\/latex]<\/p>\r\n[reveal-answer q=\"173733\"]Show Solution[\/reveal-answer][hidden-answer a=\"173733\"]\r\n\r\nNotice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.\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\nResult 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 DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>.\r\n<p style=\"padding-left: 30px;\">[latex]-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6[\/latex]<\/p>\r\n[reveal-answer q=\"173738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173738\"]\r\n\r\nNotice absolute value is not alone. Multiply both sides by the reciprocal of [latex]-\\Large\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAgain, we have a result where an absolute value is negative!\r\n\r\nThere is no solution to this equation, or DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this last video, see show more\u00a0examples of absolute value equations that have no solutions.\r\n\r\nhttps:\/\/youtu.be\/T-z5cQ58I_g\r\n\r\nWe have seen that solutions to equations can fall into three categories:\r\n<ol>\r\n \t<li>exactly one solution;<\/li>\r\n \t<li>n<span style=\"font-size: 1em;\">o solution\u00a0 (also called\u00a0<em>DNE<\/em>\u00a0<em>for\u00a0does not exist<\/em>)); or <\/span><\/li>\r\n \t<li><span style=\"font-size: 1em;\">m<\/span><span style=\"font-size: 1em;\">any solutions (also called <em>infinitely many solutions<\/em>, or we may say the solution is\u00a0<em>all real numbers).<\/em><\/span><\/li>\r\n<\/ol>\r\nKeep in mind that sometimes we do not need to do much algebra to see what the outcome will be.\r\n<div class=\"textbox shaded\">\r\n<h3>General Note: Classifying Equations<\/h3>\r\n<ul>\r\n \t<li>When an equation has a finite number of solutions (one for linear equations), the equation is called a\u00a0<strong>conditional equation<\/strong>.<\/li>\r\n \t<li>When an equation has no solution, the equation is called an\u00a0<strong>inconsistent equation<\/strong>.<\/li>\r\n \t<li>When an equation has infinitely many solutions, the equation is called an\u00a0<strong>identity<\/strong>.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve equations that have one solution, no solution, or an infinite number of solutions<\/li>\n<li>Recognize when a linear equation that contains absolute value does not have a solution<\/li>\n<li>Classify an equation as a conditional equation, an inconsistent equation, or an identity.<\/li>\n<\/ul>\n<\/div>\n<p>There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that do not have any solutions and even some that have an infinite number of solutions.<\/p>\n<h2>Equations with No Solutions<\/h2>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.<\/p>\n<p style=\"padding-left: 30px;\">[latex]12+2x\u20138=7x+5\u20135x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q790409\">Show Solution<\/span><\/p>\n<div id=\"q790409\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 <i>x<\/i> term by subtracting 2<i>x<\/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 DNE for short.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example above, a solution was not obtained. Using the the properties of equality to isolate the variable resulted instead in the false\u00a0statement [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.\u00a0Since there is no value of <i>x <\/i>that will ever make this a true statement, we say that the equation has\u00a0<i>no solution.\u00a0<\/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<div class=\"bcc-box bcc-success\">\n<h3>Think\u00a0About It<\/h3>\n<p>Try solving these equations. How many steps do you need to take before you can tell whether the equation has no solution or one solution?<\/p>\n<p>a) Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\n<p>Use the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933839\">Show Solution<\/span><\/p>\n<div id=\"q933839\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">Solve [latex]8y=3(y+4)+y[\/latex]<\/p>\n<p>First, distribute the 3 into the parentheses on the right-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]8y=3y+12+y[\/latex]<\/p>\n<p>Next, begin combining like terms.<\/p>\n<p style=\"text-align: center;\">[latex]8y=4y+12[\/latex]<\/p>\n<p>Now move the variable terms to one side. Moving the [latex]4y[\/latex] will help avoid a negative sign.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,8y=4y+12\\\\\\underline{-4y\\,\\,-4y}\\\\\\,\\,\\,\\,4y=12\\end{array}[\/latex]<\/p>\n<p>Now, divide each side by [latex]4y[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\Large\\frac{4y}{4}\\normalsize =\\Large\\frac{12}{4}\\normalsize\\\\y=3\\end{array}[\/latex]<\/p>\n<p>Because we were able to isolate <em>y<\/em> on one side and a number on the other side, we have one solution to this equation.<\/p>\n<\/div>\n<\/div>\n<p>b) Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex]<\/p>\n<p>Use the textbox below to\u00a0record how many steps you think it will take before you can tell whether there is no solution or one solution.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q937839\">Show Solution<\/span><\/p>\n<div id=\"q937839\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solve [latex]2\\left(3x-5\\right)-4x=2x+7[\/latex].<\/p>\n<p>First, distribute the 2 into the parentheses on the left-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}6x-10-4x=2x+7\\end{array}[\/latex]<\/p>\n<p>Now begin simplifying. You can combine the <em>x<\/em> terms on the left-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-10=2x+7\\end{array}[\/latex]<\/p>\n<p>Now, take a moment to ponder this equation. It\u00a0says that [latex]2x-10[\/latex] is equal to [latex]2x+7[\/latex]. Can some number times two minus 10 be equal to that same number times two plus seven?<\/p>\n<p>Pretend [latex]x=3[\/latex].<\/p>\n<p>Is it true that\u00a0[latex]2\\left(3\\right)-10=-4[\/latex] is equal to\u00a0[latex]2\\left(3\\right)+7=13[\/latex]. NO! We do not even really need to continue solving the equation, but we can just to be thorough.<\/p>\n<p>Add [latex]10[\/latex] to both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}2x-10=2x+7\\,\\,\\\\\\,\\,\\underline{+10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\2x=2x+17\\end{array}[\/latex]<\/p>\n<p>Now subtract [latex]2x[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,2x=2x+17\\\\\\,\\,\\underline{-2x\\,\\,-2x}\\\\\\,\\,\\,\\,\\,\\,\\,0=17\\end{array}[\/latex]<\/p>\n<p>We know that [latex]0\\text{ and }17[\/latex] are not equal, so there is no number that <em>x<\/em> could be to make this equation true.<\/p>\n<p>This false statement implies there are <strong>no solutions<\/strong> to this equation, or DNE (does not exist) for short.<\/p>\n<\/div>\n<\/div>\n<\/div>\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 <i>x<\/i>. It is possible to have an equation where any value for <em>x<\/em> will provide a solution to the equation. In the example below, notice how combining the terms [latex]5x[\/latex] and [latex]-4x[\/latex] on the left\u00a0leaves us with an equation with exactly the same terms on both sides of the equal sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.<\/p>\n<p style=\"padding-left: 30px;\">[latex]5x+3\u20134x=3+x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q773733\">Show Solution<\/span><\/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 <i>x<\/i> term by subtracting <i>x<\/i> 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;<\/p>\n<\/div>\n<\/div>\n<\/div>\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]\u00a0into 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 x will make the original statement true.<\/p>\n<p>Indeed, after combining like terms, the equation [latex]x+3=3+x[\/latex] was obtained. It is certainly true that the quantity [latex]x[\/latex] with [latex]3[\/latex] added to it is equal to [latex]3[\/latex] with [latex]x[\/latex] added to it by the commutative property of addition.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.<\/p>\n<p style=\"padding-left: 30px;\">[latex]3\\left(2x-5\\right)=6x-15[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q973733\">Show Solution<\/span><\/p>\n<div id=\"q973733\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute the [latex]3[\/latex] through the parentheses on the left-hand side.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3\\left(2x-5\\right)=6x-15\\\\6x-15=6x-15\\end{array}[\/latex]<\/p>\n<p>Wait! This looks just like the previous example. You have the same expression on both sides of an equal sign. \u00a0No matter what number you choose for <em>x<\/em>, you will have a true statement. We can finish the algebra:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6x-15=6x-15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-6x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6x\\,}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-15\\,\\,=\\,\\,-15\\end{array}[\/latex]<\/p>\n<p>This true statement implies there are an infinite number of solutions to this equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video for demonstrations of equations with <em>no solutions<\/em> and\u00a0<em>infinitely many solutions<\/em>.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Linear Equations with No Solutions or Infinite Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/iLkZ3o4wVxU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The next video demonstrates equations with no or infinitely many solutions involving parentheses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Linear Equations with No Solutions of Infinite Solutions (Parentheses)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/EU_NEo1QBJ0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><span style=\"color: #ff9900;\">[Optional]<\/span> Absolute Value Equations with No Solutions<\/h2>\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\u00a0 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 (DNE, for short). Notice how this happens in the next two examples.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.<\/p>\n<p style=\"padding-left: 30px;\">[latex]7+\\left|2x-5\\right|=4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173733\">Show Solution<\/span><\/p>\n<div id=\"q173733\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>.<\/p>\n<p style=\"padding-left: 30px;\">[latex]-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q173738\">Show Solution<\/span><\/p>\n<div id=\"q173738\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice absolute value is not alone. Multiply both sides by the reciprocal of [latex]-\\Large\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\Large\\frac{1}{2}\\normalsize\\left|x+3\\right|=\\left(-2\\right)6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left|x+3\\right|=-12\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Again, we have a result where an absolute value is negative!<\/p>\n<p>There is no solution to this equation, or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this last video, see show more\u00a0examples of absolute value equations that have no solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Absolute Value Equations with No Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/T-z5cQ58I_g?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We have seen that solutions to equations can fall into three categories:<\/p>\n<ol>\n<li>exactly one solution;<\/li>\n<li>n<span style=\"font-size: 1em;\">o solution\u00a0 (also called\u00a0<em>DNE<\/em>\u00a0<em>for\u00a0does not exist<\/em>)); or <\/span><\/li>\n<li><span style=\"font-size: 1em;\">m<\/span><span style=\"font-size: 1em;\">any solutions (also called <em>infinitely many solutions<\/em>, or we may say the solution is\u00a0<em>all real numbers).<\/em><\/span><\/li>\n<\/ol>\n<p>Keep in mind that sometimes we do not need to do much algebra to see what the outcome will be.<\/p>\n<div class=\"textbox shaded\">\n<h3>General Note: Classifying Equations<\/h3>\n<ul>\n<li>When an equation has a finite number of solutions (one for linear equations), the equation is called a\u00a0<strong>conditional equation<\/strong>.<\/li>\n<li>When an equation has no solution, the equation is called an\u00a0<strong>inconsistent equation<\/strong>.<\/li>\n<li>When an equation has infinitely many solutions, the equation is called an\u00a0<strong>identity<\/strong>.<\/li>\n<\/ul>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-79\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Absolute Value Equations with No Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/T-z5cQ58I_g\">https:\/\/youtu.be\/T-z5cQ58I_g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Linear Equations with No Solutions or Infinite Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/iLkZ3o4wVxU\">https:\/\/youtu.be\/iLkZ3o4wVxU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Linear Equations with No Solutions of Infinite Solutions (Parentheses). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/EU_NEo1QBJ0\">https:\/\/youtu.be\/EU_NEo1QBJ0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Absolute Value Equations with No Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/T-z5cQ58I_g\">https:\/\/youtu.be\/T-z5cQ58I_g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>GENERAL NOTE: CLASSIFYING EQUATIONS. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University . <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Beginning and Intermediate Algebra. <strong>Authored by<\/strong>: Tyler Wallace. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/wallace.ccfaculty.org\/book\/book.html\">http:\/\/wallace.ccfaculty.org\/book\/book.html<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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