{"id":6724,"date":"2020-10-08T16:07:01","date_gmt":"2020-10-08T16:07:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6724"},"modified":"2026-02-24T21:11:42","modified_gmt":"2026-02-24T21:11:42","slug":"2-5-absolute-value-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/2-5-absolute-value-equations\/","title":{"raw":"2.5: Absolute Value Equations","rendered":"2.5: Absolute Value Equations"},"content":{"raw":"
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section 2.5 Learning Objectives<\/h3>\r\n2.5:\u00a0 Absolute Value Equations<\/strong>\r\n
    \r\n \t
  • Solve absolute value equations<\/li>\r\n \t
  • Recognize when an absolute value equation has no solution<\/li>\r\n \t
  • Solve absolute value equations containing two absolute values<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nThe absolute value<\/b> of a number or expression describes its distance from 0 on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or 0.\r\n\r\nFor example, [latex]\u22124[\/latex] and 4 both have an absolute value of 4 because they are each 4 units from 0 on a number line\u2014though they are located in opposite directions from 0 on the number line.\r\n

    Solving equations containing absolute values<\/h2>\r\n
    \r\n

    Key Takeaways<\/h3>\r\nWhen solving absolute value equations<\/b> and inequalities<\/b>, you have to consider both the behavior of absolute value and the properties of equality and inequality. Because both positive and negative values have a positive absolute value, solving absolute value equations or inequalities means finding the solution for both the positive and the negative values.<\/strong>\r\n\r\n \r\n

    REMEMBER: We must always set up two cases when solving absolute value functions; one positive case and one negative case<\/strong><\/h2>\r\n<\/div>\r\nLet\u2019s first look at a very basic example.\r\n

    [latex] \\displaystyle \\left| x \\right|=5[\/latex]<\/p>\r\nThis equation is read \u201cthe absolute value of x <\/i>is equal to five.\u201d The solution is the value(s) that are five units away from 0 on a number line.\r\n\r\nYou might think of 5 right away; that is one solution to the equation. Notice that [latex]\u22125[\/latex] is also a solution because [latex]\u22125[\/latex] is 5 units away from 0 in the opposite direction. So, the solution to this equation [latex] \\displaystyle \\left| x \\right|=5[\/latex] is [latex]x = \u22125[\/latex] or [latex]x = 5[\/latex].\r\n

    \r\n

    Solving Equations of the Form [latex]|x|=a[\/latex]<\/h3>\r\nFor any positive number a<\/i>, the solution of [latex]\\left|x\\right|=a[\/latex] is\r\n

    [latex]x=a[\/latex] or [latex]x=\u2212a[\/latex]<\/p>\r\nx <\/i>can be a single variable or any algebraic expression.\r\n\r\n<\/div>\r\nYou can solve a more complex absolute value problem in a similar fashion.\r\n

    \r\n

    Example 1<\/h3>\r\nSolve for x<\/em>: [latex]\\hspace{.05in}\\displaystyle \\left| x+5\\right|=15[\/latex]\r\n[reveal-answer q=\"624457\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624457\"]\r\n\r\nThis equation asks you to find what number plus 5 has an absolute value of 15. Since 15 and [latex]\u221215[\/latex] both have an absolute value of 15, the absolute value equation is true when the quantity [latex]x + 5[\/latex] is 15 or<\/i> [latex]x + 5[\/latex] is [latex]\u221215[\/latex], since [latex]|15|=15[\/latex] and [latex]|\u221215|=15[\/latex]. So, you need to find out what value for x<\/i> will make this expression equal to 15 as well as what value for x<\/em> will make the expression equal to [latex]\u221215[\/latex]. Solving the two equations you get\r\n

    [latex] \\displaystyle \\begin{array}{l}x+5=15\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,{x+5=-15}\\\\\\underline{\\,\\,\\,\\,\\,-5\\,\\,\\,\\,-5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,10\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,=-20\\end{array}[\/latex]<\/p>\r\nYou can check these two solutions in the absolute value equation to see if [latex]x=10[\/latex] and [latex]x=\u221220[\/latex] are correct.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\left| x+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+5 \\right|=15\\\\\\left| 10+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -20+5 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 15 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -15 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nThe following video provides worked examples of solving linear equations with absolute value terms.\r\nhttps:\/\/youtu.be\/U-7fF-W8_xE\r\n

    \r\n

    Example 2<\/h3>\r\nSolve for\u00a0x<\/em>: [latex]\\hspace{.05in}\\displaystyle \\left| 2x\\right|=6[\/latex]\r\n[reveal-answer q=\"624455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624455\"]\r\n\r\nThis equation asks you to find what number times 2 has an absolute value of 6.\r\n\r\nSince 6 and [latex]\u22126[\/latex] both have an absolute value of 6, the absolute value equation is true when the quantity [latex]2x[\/latex] is 6 or<\/i> [latex]2x[\/latex] is [latex]\u22126[\/latex], since [latex]|6|=6[\/latex] and [latex]|\u22126|=6[\/latex].\r\n\r\nSo, you need to find out what value for x<\/i> will make this expression equal to 6 as well as what value for x<\/em> will make the expression equal to [latex]\u22126[\/latex].\r\n\r\nSolving the two equations you get\r\n

    [latex]2x=6\\text{ or }2x=-6[\/latex]<\/p>\r\n

    [latex]\\frac{2x}{2}=\\frac{6}{2}\\text{ or }\\frac{2x}{2}=\\frac{-6}{2}[\/latex]<\/p>\r\n

    [latex]x=3\\text{ or }x=-3[\/latex]<\/p>\r\nYou can check these two solutions in the absolute value equation to see if [latex]x=3[\/latex] and [latex]x=\u22123[\/latex] are correct.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\left|3\\cdot2 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -3\\cdot2 \\right|=6\\\\\\left| 6 \\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -6 \\right|=6\\\\\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n

    \r\n

    Example 3<\/h3>\r\nSolve for\u00a0k<\/em>: [latex]\\hspace{.05in}\\displaystyle\\frac{1}{3}\\left|k\\right|=12[\/latex]\r\n[reveal-answer q=\"604455\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604455\"]\r\n\r\nNotice how this example is different from the last; [latex] \\displaystyle\\frac{1}{3}[\/latex] is outside the absolute value grouping symbols. This means we need to isolate the absolute value first, then apply the definition of absolute value.\r\n\r\nFirst, isolate the absolute value term by multiplying by the inverse of [latex] \\displaystyle\\frac{1}{3}[\/latex]:\r\n

    [latex]\\begin{array}{r}\\frac{1}{3}\\left|k\\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\left(3\\right)\\frac{1}{3}\\left|k\\right|=\\left(3\\right)12\\\\\\left|k\\right|=36\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nApply the definition of absolute value:\r\n\r\n[latex] \\displaystyle{k }=36\\text{ or }{k }=-36[\/latex]\r\n\r\nYou can check these two solutions in the absolute value equation to see if [latex]x=36[\/latex] and [latex]x=\u221236[\/latex] are correct.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\frac{1}{3}\\left|36 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{1}{3}\\left|-36 \\right|=12\\\\\\left| 12 \\right|=12\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -12 \\right|=12\\\\\\end{array}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the following video you will see two examples of how to solve an absolute value equation, one with integers and one with fractions.\r\nhttps:\/\/youtu.be\/CTLnJ955xzc\r\n

    \r\n

    Example 4<\/h3>\r\nSolve for p<\/i>:\u00a0[latex]\\hspace{.05in}\\left|2p\u20134\\right|=26[\/latex]\r\n\r\n[reveal-answer q=\"371950\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"371950\"]\r\n\r\nWrite the two equations that will give an absolute value of 26.\r\n

    [latex] \\displaystyle 2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,2p-4=\\,-26[\/latex]<\/p>\r\nSolve each equation for p <\/i>by isolating the variable.<\/i>\r\n

    [latex] \\displaystyle \\begin{array}{r}2p-4=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2p-4=\\,-26\\\\\\underline{\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,+4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,+4\\,\\,\\,\\,\\,\\,\\,+4}\\\\\\underline{2p}\\,\\,\\,\\,\\,\\,=\\underline{30}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{2p}\\,\\,\\,\\,\\,=\\,\\underline{-22}\\\\2\\,\\,\\,\\,\\,\\,\\,=\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,\\,\\,\\,\\,\\,2\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,p=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,p=\\,-11\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2p-4 \\right|=26\\\\\\left| 2(15)-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\left| 2(-11)-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\left| 30-4 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -22-4 \\right|=26\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 26 \\right|=26\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -26 \\right|=26\\end{array}[\/latex]<\/p>\r\nBoth solutions check!\r\n

    Answer<\/h4>\r\n[latex]p=15[\/latex] or [latex]p=-11[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, we show more examples of solving multi-step absolute value equations.\r\n\r\nhttps:\/\/youtu.be\/4g-o_-mAFpc\r\n\r\nNow let's look at an example where you need to do an algebraic step or two before you can write your two equations. The goal here is to get the absolute value on one side of the equation by itself. Then we can proceed as we did in the previous example.\r\n
    \r\n

    Example 5<\/h3>\r\nSolve for w<\/i>:\u00a0[latex]\\hspace{.05in}3\\left|4w\u20131\\right|\u20135=10[\/latex]\r\n\r\n[reveal-answer q=\"303228\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303228\"]\r\n\r\nIsolate the term with the absolute value by adding 5 to both sides.\r\n

    [latex]\\begin{array}{r}3\\left|4w-1\\right|-5=10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+5\\,\\,\\,+5}\\\\ 3\\left|4w-1\\right|=15\\end{array}[\/latex]<\/p>\r\nDivide both sides by 3.\u00a0Now the absolute value is isolated.\r\n

    [latex]\\begin{array}{r} \\underline{3\\left|4w-1\\right|}=\\underline{15}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\,\\\\\\left|4w-1\\right|=\\,\\,5\\end{array}[\/latex]<\/p>\r\nWrite the two equations that will give an absolute value of 5 and solve them.\r\n

    [latex] \\displaystyle \\begin{array}{r}4w-1=5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4w-1=-5\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+1\\,\\,+1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,\\,+1}\\\\\\,\\,\\,\\,\\,\\underline{4w}=\\underline{6}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{4w}\\,\\,\\,\\,\\,\\,\\,=\\underline{-4}\\\\4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=-1\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,w=\\frac{3}{2}\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\3\\left| 4\\left( \\frac{3}{2} \\right)-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 4w-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,3\\left| \\frac{12}{2}-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,3\\left| 4(-1)-1\\, \\right|-5=10\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,3\\left| 6-1\\, \\right|-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -4-1\\, \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left(5\\right)-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\left| -5 \\right|-5=10\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15-5=10\\\\10=10\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,10=10\\end{array}[\/latex]<\/p>\r\nBoth solutions check\r\n

    Answer<\/h4>\r\n[latex]w=-1\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,w=\\frac{3}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nIn the two videos that follow, we show examples of how to solve an absolute value equation that requires you to isolate the absolute value first using mathematical operations.\r\n\r\nhttps:\/\/youtu.be\/-HrOMkIiSfU\r\n\r\nhttps:\/\/youtu.be\/2bEA7HoDfpk\r\n

    <\/h2>\r\n

    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 from 0 on a number line, so it must be a positive number. When an absolute value expression is equal to a negative number, we say the equation has no solution, or DNE. Notice how this happens in the next two examples.\r\n
    \r\n

    Example 6<\/h3>\r\nSolve for x<\/i>:\u00a0[latex]\\hspace{.05in}7+\\left|2x-5\\right|=4[\/latex]\r\n\r\n[reveal-answer q=\"173733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173733\"]Notice absolute value is not alone. Subtract [latex]7[\/latex] from each side to isolate the absolute value.\r\n

    [latex]\\begin{array}{r}7+\\left|2x-5\\right|=4\\,\\,\\,\\,\\\\\\underline{\\,-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,}\\\\\\left|2x-5\\right|=-3\\end{array}[\/latex]<\/p>\r\nThe result of absolute value is negative! The result of an absolute value must always be nonnegative, 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

    \r\n

    Example 7<\/h3>\r\nSolve for x<\/i>:\u00a0[latex]\\hspace{.05in}-\\frac{1}{2}\\left|x+3\\right|=6[\/latex]\r\n\r\n[reveal-answer q=\"173738\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173738\"]Notice absolute value is not alone, multiply both sides by the reciprocal of [latex]-\\frac{1}{2}[\/latex], which is [latex]-2[\/latex].\r\n

    [latex]\\begin{array}{r}-\\frac{1}{2}\\left|x+3\\right|=6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\left(-2\\right)-\\frac{1}{2}\\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, we show more\u00a0examples of absolute value equations that have no solutions.\r\n\r\nhttps:\/\/youtu.be\/T-z5cQ58I_g\r\n

    <\/h2>\r\n

    Solving equations with two absolute values<\/h2>\r\nIf we are given an equality of two absolute value expressions, for example [latex]|3x+4|=|2x-1|[\/latex], we apply the same idea.\u00a0 The equation will be true when the two expressions inside the absolute values are the same distance from zero, which occurs if they have the same value or are opposites.\u00a0 The expressions will have the same value when they are equal to each other.\u00a0 In our example, this implies [latex]3x+4=2x-1[\/latex].\u00a0 The second equation will be obtained by taking the opposite of either expression.\u00a0 Be careful to distribute the negative, which we can indicate with parentheses.\u00a0 Here, this would be given by [latex]3x+4=-(2x-1)[\/latex].\u00a0 See the full solution below.\r\n
    \r\n

    Example 8<\/h3>\r\nSolve for\u00a0x<\/em>: [latex] \\hspace{.05in}|3x+4|=|2x-1|[\/latex]\r\n\r\n[reveal-answer q=\"322432\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322432\"]The values for x <\/em>that will satisfy the equation will be those that result in the expressions inside the absolute values being equal to each other or opposites of each other.\r\n\r\n

    [latex]\r\n\\begin{array}{rclcrcl}\r\n3x+4 &=& 2x-1 & \\quad\\text{or}\\quad & 3x+4 &=& -(2x-1) \\\\\r\n\\underline{-2x\\phantom{{}+4}} & & \\underline{-2x\\phantom{{}-1}} & & 3x+4 &=& -2x+1 \\\\\r\nx+4 &=& -1 & & \\underline{+2x\\phantom{{}+4}} & & \\underline{+2x\\phantom{{}+1}} \\\\\r\n\\underline{\\phantom{x+}{-4}} & & \\underline{\\phantom{2x}{-4}} & & 5x+4 &=& 1 \\\\\r\nx &=& -5 & & \\underline{\\phantom{5x+}{-4}} & & \\underline{\\phantom{+1}{-4}} \\\\\r\n& & & & 5x &=& -3 \\\\\r\n& & & & x &=& -\\dfrac{3}{5}\r\n\\end{array}\r\n[\/latex]<\/p>\r\n\r\nCheck the solutions in the original equation.\r\n\r\n[latex] \\hspace{1.5in} \\begin{array}{rclcrcl} |3(-5)+4|&=&|2(-5)-1|& \\hspace{1in} & \\left|3\\left(-\\frac{3}{5}\\right)+4\\right|&=&\\left|2\\left(-\\frac{3}{5}\\right)-1\\right| \\\\ |-15+4|&=&|-10-1|& & \\left|-\\frac{9}{5}+4\\right|&=&\\left|-\\frac{6}{5}-1\\right| \\\\ |-11|&=&|-11| & & \\left| \\frac{11}{5}\\right|&=&\\left|-\\frac{11}{5}\\right| \\\\ 11&=&11 & & \\frac{11}{5}&=&\\frac{11}{5} \\end{array} [\/latex]\r\n\r\nBoth solutions check.\r\n

    Answer<\/span><\/h4>\r\n[latex]x=-5[\/latex] or [latex]x=-\\frac{3}{5}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nThe video below shows some more examples of solving an equation with two absolute values.\r\n\r\nhttps:\/\/youtu.be\/g_bDUvVgTB0\r\n

    Summary<\/h2>\r\nEquations are mathematical statements that combine two expressions of equal value. An algebraic equation can be solved by isolating the variable on one side of the equation using the properties of equality. To check the solution of an algebraic equation, substitute the value of the variable into the original equation.\r\n\r\nComplex, multi-step equations often require multi-step solutions. Before you can begin to isolate a variable, you may need to simplify the equation first.\u00a0 If your multi-step equation has an absolute value, you will need to solve two equations,<\/strong> sometimes isolating the absolute value expression first.","rendered":"
    \n

    section 2.5 Learning Objectives<\/h3>\n

    2.5:\u00a0 Absolute Value Equations<\/strong><\/p>\n

      \n
    • Solve absolute value equations<\/li>\n
    • Recognize when an absolute value equation has no solution<\/li>\n
    • Solve absolute value equations containing two absolute values<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      The absolute value<\/b> of a number or expression describes its distance from 0 on a number line. Since the absolute value expresses only the distance, not the direction of the number on a number line, it is always expressed as a positive number or 0.<\/p>\n

      For example, [latex]\u22124[\/latex] and 4 both have an absolute value of 4 because they are each 4 units from 0 on a number line\u2014though they are located in opposite directions from 0 on the number line.<\/p>\n

      Solving equations containing absolute values<\/h2>\n
      \n

      Key Takeaways<\/h3>\n

      When solving absolute value equations<\/b> and inequalities<\/b>, you have to consider both the behavior of absolute value and the properties of equality and inequality. Because both positive and negative values have a positive absolute value, solving absolute value equations or inequalities means finding the solution for both the positive and the negative values.<\/strong><\/p>\n

       <\/p>\n

      REMEMBER: We must always set up two cases when solving absolute value functions; one positive case and one negative case<\/strong><\/h2>\n<\/div>\n

      Let\u2019s first look at a very basic example.<\/p>\n

      [latex]\\displaystyle \\left| x \\right|=5[\/latex]<\/p>\n

      This equation is read \u201cthe absolute value of x <\/i>is equal to five.\u201d The solution is the value(s) that are five units away from 0 on a number line.<\/p>\n

      You might think of 5 right away; that is one solution to the equation. Notice that [latex]\u22125[\/latex] is also a solution because [latex]\u22125[\/latex] is 5 units away from 0 in the opposite direction. So, the solution to this equation [latex]\\displaystyle \\left| x \\right|=5[\/latex] is [latex]x = \u22125[\/latex] or [latex]x = 5[\/latex].<\/p>\n

      \n

      Solving Equations of the Form [latex]|x|=a[\/latex]<\/h3>\n

      For any positive number a<\/i>, the solution of [latex]\\left|x\\right|=a[\/latex] is<\/p>\n

      [latex]x=a[\/latex] or [latex]x=\u2212a[\/latex]<\/p>\n

      x <\/i>can be a single variable or any algebraic expression.<\/p>\n<\/div>\n

      You can solve a more complex absolute value problem in a similar fashion.<\/p>\n

      \n

      Example 1<\/h3>\n

      Solve for x<\/em>: [latex]\\hspace{.05in}\\displaystyle \\left| x+5\\right|=15[\/latex]<\/p>\n

      Show Solution<\/span><\/p>\n
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      This equation asks you to find what number plus 5 has an absolute value of 15. Since 15 and [latex]\u221215[\/latex] both have an absolute value of 15, the absolute value equation is true when the quantity [latex]x + 5[\/latex] is 15 or<\/i> [latex]x + 5[\/latex] is [latex]\u221215[\/latex], since [latex]|15|=15[\/latex] and [latex]|\u221215|=15[\/latex]. So, you need to find out what value for x<\/i> will make this expression equal to 15 as well as what value for x<\/em> will make the expression equal to [latex]\u221215[\/latex]. Solving the two equations you get<\/p>\n

      [latex]\\displaystyle \\begin{array}{l}x+5=15\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,{x+5=-15}\\\\\\underline{\\,\\,\\,\\,\\,-5\\,\\,\\,\\,-5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{-5\\,\\,\\,\\,\\,\\,\\,\\,\\,-5}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,10\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,=-20\\end{array}[\/latex]<\/p>\n

      You can check these two solutions in the absolute value equation to see if [latex]x=10[\/latex] and [latex]x=\u221220[\/latex] are correct.<\/p>\n

      [latex]\\displaystyle \\begin{array}{r}\\,\\,\\left| x+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+5 \\right|=15\\\\\\left| 10+5 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -20+5 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 15 \\right|=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -15 \\right|=15\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,15=15\\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n

      The following video provides worked examples of solving linear equations with absolute value terms.
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