{"id":6730,"date":"2020-10-08T16:25:13","date_gmt":"2020-10-08T16:25:13","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6730"},"modified":"2026-02-24T21:20:27","modified_gmt":"2026-02-24T21:20:27","slug":"2-6-absolute-value-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/2-6-absolute-value-inequalities\/","title":{"raw":"2.6: Absolute Value Inequalities","rendered":"2.6: Absolute Value Inequalities"},"content":{"raw":"
\r\n

section 2.6 Learning Objectives<\/h3>\r\n2.6:\u00a0 Absolute Value Inequalities<\/strong>\r\n
    \r\n \t
  • Determine whether an absolute value inequality corresponds to a union or an intersection of inequalities<\/li>\r\n \t
  • Solve absolute value inequalities and express the solutions graphically and in interval notation<\/li>\r\n \t
  • Recognize when an absolute value inequality has no solution or all real numbers as the solution<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n

    Solve Inequalities Containing Absolute Value<\/h2>\r\nLet us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us start with a simple inequality.\r\n

    [latex]\\left|x\\right|\\leq 4[\/latex]<\/p>\r\nThis inequality is read, \u201cthe absolute value of x <\/i>is less than or equal to\u00a0[latex]4[\/latex].\u201d If you are asked to solve for x<\/i>, you want to find out what values of x <\/i>are\u00a0[latex]4[\/latex] units or less away from\u00a0[latex]0[\/latex] on a number line. You could start by thinking about the number line and what values of x <\/i>would satisfy this equation.\r\n\r\n[latex]4[\/latex] and [latex]\u22124[\/latex] are both four units away from\u00a0[latex]0[\/latex], so they are solutions.\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are also solutions because each of these values is less than [latex]4[\/latex] units away from\u00a0[latex]0[\/latex]. So are\u00a0[latex]1[\/latex] and [latex]\u22121[\/latex],[latex]0.5[\/latex] and [latex]\u22120.5[\/latex], and so on\u2014there are an infinite number of values for x<\/i> that will satisfy this inequality.\r\n\r\nThe graph of this inequality will have two closed circles, at\u00a0[latex]4[\/latex] and [latex]\u22124[\/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.\r\n\r\n\"Number\r\n\r\nThe solution can be written this way:\r\n\r\nInequality notation: [latex]-4\\leq x\\leq4[\/latex]\r\n\r\nInterval notation: [latex]\\left[-4,4\\right][\/latex]\r\n\r\nThe situation is a little different when the inequality sign is \u201cgreater than\u201d or \u201cgreater than or equal to.\u201d Consider the simple inequality [latex]\\left|x\\right|>3[\/latex]. Again, you could think of the number line and what values of x<\/i> are greater than\u00a0[latex]3[\/latex] units away from zero. This time,\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values.\u00a0[latex]2[\/latex] and [latex]\u22122[\/latex] would not be solutions because they are not more than [latex]3[\/latex] units away from\u00a0[latex]0[\/latex]. But\u00a0[latex]5[\/latex] and [latex]\u22125[\/latex] would work and so would all of the values extending to the left of [latex]\u22123[\/latex] and to the right of\u00a0[latex]3[\/latex]. The graph would look like the one below.\r\n\r\n\"Number\r\n\r\nThe solution to this inequality can be written this way:\r\n\r\nInequality notation:<\/i> [latex]x<\u22123[\/latex] or [latex]x>3[\/latex].\r\n\r\nInterval notation: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]\r\n\r\nIn the following video, you will see examples of how to solve and express the solution\u00a0to absolute value inequalities involving both and<\/em> and or<\/em>.\r\n\r\nhttps:\/\/youtu.be\/0cXxATY2S-k\r\n

    Writing Solutions to Absolute Value Inequalities<\/h3>\r\nFor any positive value of a\u00a0<\/i>and\u00a0x,<\/em>\u00a0a single variable, or any algebraic expression:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Absolute Value Inequality<\/strong><\/td>\r\nEquivalent Inequality<\/strong><\/td>\r\nInterval Notation<\/strong><\/td>\r\n<\/tr>\r\n
    [latex]\\left|{ x }\\right|\\le{ a}[\/latex]<\/td>\r\n[latex]{ -a}\\le{x}\\le{ a} \\\\ \\mbox{Equivalently: } x\\le a \\mbox{ and } x\\ge -a [\/latex]<\/td>\r\n[latex]\\left[-a, a\\right][\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\left| x \\right|\\lt{a}[\/latex]<\/td>\r\n[latex]{ -a}\\lt{x}\\lt{ a} \\\\ \\mbox{Equivalently: } x\\lt a \\mbox{ and } x\\gt -a [\/latex]<\/td>\r\n[latex]\\left(-a, a\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\left| x \\right|\\ge{ a}[\/latex]<\/td>\r\n[latex]{x}\\le\\text{\u2212a}[\/latex] or [latex]{x}\\ge{ a}[\/latex]<\/td>\r\n\u00a0[latex]\\left(-\\infty,-a\\right]\\cup\\left[a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\\left| x \\right|\\gt\\text{a}[\/latex]<\/td>\r\n[latex]\\displaystyle{x}\\lt\\text{\u2212a}[\/latex]\u00a0or [latex]{x}\\gt{ a}[\/latex]<\/td>\r\n\u00a0[latex]\\left(-\\infty,-a\\right)\\cup\\left(a,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLet us look at a few more examples of inequalities containing absolute value.\r\n
    \r\n

    Example 2<\/h3>\r\nSolve for x<\/i>.\r\n\r\n[latex]\\left|x+3\\right|\\gt4[\/latex]\r\n\r\n[reveal-answer q=\"867809\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867809\"]\r\n\r\nSince this is a \u201cgreater than\u201d inequality, the solution can be rewritten according to the \u201cgreater than\u201d rule.\r\n

    [latex] \\displaystyle x+3<-4\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,\\,x+3>4[\/latex]<\/p>\r\nSolve each inequality.\r\n

    [latex]\\begin{array}{r}x+3<-4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x+3>4\\\\\\underline{\\,\\,\\,\\,-3\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-3\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,<-7\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,\\,\\,>1\\\\\\\\x<-7\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x>1\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck the solutions in the original equation to be sure they work. Check the end point of the first related equation, [latex]\u22127[\/latex] and the end point of the second related equation, 1.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\left| x+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\\\\\left| -7+3 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 1+3 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\left| -4 \\right|=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 4 \\right|=4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4=4\\end{array}[\/latex]<\/p>\r\nTry [latex]\u221210[\/latex], a value less than [latex]\u22127[\/latex], and 5, a value greater than 1, to check the inequality.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| x+3 \\right|>4\\\\\\left| -10+3 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 5+3 \\right|>4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| -7 \\right|>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 8 \\right|>4\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,7>4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,8>4\\end{array}[\/latex]<\/p>\r\nBoth solutions check!\r\n\r\nInequality notation: [latex] \\displaystyle x<-7\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,x>1[\/latex]\r\n\r\nInterval notation: [latex]\\left(-\\infty, -7\\right)\\cup\\left(1,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n

    \"Number<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    \r\n

    Example 2<\/h3>\r\nSolve for y.\u00a0<\/i>\r\n\r\n[latex] 3\\left| 2y+6 \\right|-9<27[\/latex]\r\n\r\n[reveal-answer q=\"632256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"632256\"]\r\n\r\nBegin isolating the absolute value by adding 9 to both sides of the inequality.\r\n

    [latex] \\displaystyle \\begin{array}{r}3\\left| 2y+6 \\right|-9<27\\\\\\underline{\\,\\,+9\\,\\,\\,+9}\\\\3\\left| 2y+6 \\right|\\,\\,\\,\\,\\,\\,\\,\\,<36\\end{array}[\/latex]<\/p>\r\nDivide both sides by 3 to isolate the absolute value.\r\n

    [latex]\\begin{array}{r}\\underline{3\\left| 2y+6 \\right|}\\,<\\underline{36}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2y+6 \\right|<12\\end{array}[\/latex]<\/p>\r\nWrite the absolute value inequality using the \u201cless than\u201d rule.\u00a0Subtract 6 from each part of the inequality.\r\n

    [latex]\\begin{array}{r}-12<2y+6<12\\\\\\underline{\\,\\,-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\,\\,\\,-6}\\\\-18\\,<\\,2y\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,6\\,\\end{array}[\/latex]<\/p>\r\nDivide by\u00a0[latex]2[\/latex] to isolate the variable.\r\n

    [latex]\\begin{array}{r}\\underline{-18}<\\underline{2y}<\\underline{\\,6\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\-9<\\,\\,y\\,\\,\\,\\,<\\,3\\end{array}[\/latex]<\/p>\r\n\r\n

    Answer<\/h4>\r\nInequality notation: [latex] \\displaystyle -9<\\,\\,y\\,\\,<3[\/latex]\r\n\r\nInterval notation: [latex]\\left(-9,3\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn section 2.4, we saw that tripartite inequalities could also be solved by splitting the problem into a compound inequality adjoined by and<\/em>.\u00a0 This strategy can also be utilized here, providing an approach more similar to the \"greater than\" inequalities.\u00a0 Below, you will find the previous example redone with this approach.\r\n
    \r\n

    Example 3<\/h3>\r\nSolve for y.\u00a0<\/i>\r\n\r\n[latex]3\\left| 2y+6 \\right|-9<27[\/latex]\r\n\r\n[reveal-answer q=\"717249\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"717249\"]\r\n\r\nBegin isolating the absolute value by adding 9 to both sides of the inequality.\r\n

    [latex] \\displaystyle \\begin{array}{r}3\\left| 2y+6 \\right|-9<27\\\\\\underline{\\,\\,+9\\,\\,\\,+9}\\\\3\\left| 2y+6 \\right|\\,\\,\\,\\,\\,\\,\\,\\,<36\\end{array}[\/latex]<\/p>\r\nDivide both sides by 3 to isolate the absolute value.\r\n

    [latex]\\begin{array}{r}\\underline{3\\left| 2y+6 \\right|}\\,<\\underline{36}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left| 2y+6 \\right|<12\\end{array}[\/latex]<\/p>\r\nInstead of creating a tripartite inequality, this time we use the equivalent\u00a0and<\/em> compound inequality.\r\n

    [latex]\r\n\\begin{array}{rclcrcl}\r\n2y+6 &<& 12 & \\quad\\text{and}\\quad & 2y+6 &>& -12 \\\\\r\n\\underline{\\phantom{2y+}{-6}} & & \\underline{-6\\phantom{2}} & & \\underline{\\phantom{2y+}{-6}} & & \\underline{-6\\phantom{-12}} \\\\\r\n\\dfrac{2y}{2} &<& \\dfrac{6}{2} & & \\dfrac{2y}{2} &>& \\dfrac{-18}{2} \\\\[6pt]\r\ny &<& 3 & \\quad\\text{and}\\quad & y &>& -9\r\n\\end{array}\r\n[\/latex]<\/p>\r\nWe want the intersection of the two regions, which corresponds to all values that are simultaneously less than [latex]3[\/latex] and greater than [latex]-9[\/latex], which can be written as\r\n

    [latex]-9<\\,\\,y\\,\\,<3[\/latex]<\/p>\r\nThis matches the answer we obtained when solving this directly as a tripartite inequality.\r\n

    Answer<\/h4>\r\nInequality notation: [latex] \\displaystyle -9<\\,\\,y\\,\\,<3[\/latex]\r\n\r\nInterval notation: [latex]\\left(-9,3\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n\r\nIn the following video, you will see an example of solving multi-step absolute value inequalities involving an and<\/em>\u00a0situation.\r\n\r\n[embed]https:\/\/youtu.be\/d-hUviSkmqE[\/embed]\r\n\r\nThe next video gives another example of an and<\/em> situation.\r\n\r\n[embed]https:\/\/youtu.be\/ttUaRf-GzpM[\/embed]\r\n\r\nIn the last video that follows, you will see an example of an or<\/em> situation\u00a0where you need to isolate the absolute value first.\r\n\r\n[embed]https:\/\/youtu.be\/5jRUuiMUxWQ[\/embed]\r\n

    Identify Cases of Inequalities Containing Absolute Value That Have No Solutions<\/h3>\r\nAs with equations, there may be instances where there is no solution to an inequality.\u00a0 This occurs if we obtain a statement that implies an absolute value is less than a negative number.\u00a0 No value for the variable can ever make this true since absolute values are always nonnegative.\r\n
    \r\n

    Example 4<\/h3>\r\nSolve for x<\/i>: [latex]\\hspace{.05in}\\left|2x+3\\right|+9\\leq 7[\/latex]\r\n\r\n[reveal-answer q=\"931656\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"931656\"]\r\n\r\nIsolate the absolute value by subtracting\u00a0[latex]9[\/latex] from both sides of the inequality.\r\n

    [latex] \\displaystyle \\begin{array}{r}\\left| 2x+3 \\right|+9\\,\\le \\,\\,\\,7\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-9\\,\\,\\,\\,\\,-9}\\\\\\,\\,\\,\\,\\,\\,\\,\\left| 2x+3 \\right|\\,\\,\\,\\le -2\\,\\end{array}[\/latex]<\/p>\r\nThe absolute value of a quantity can never be a negative number and therefore never less than a negative number.\u00a0 So there is no solution to the inequality.\r\n

    Answer<\/span><\/h4>\r\nNo Solution\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

    Identify Cases of Inequalities Containing Absolute Value That Have All Real Numbers as the Solution<\/h3>\r\nSince absolute values are always nonnegative, if we instead arrive at a statement that implies an absolute value is greater than a negative number, this will hold true for every value of the variable.\r\n
    \r\n

    Example 5<\/h3>\r\nSolve for\u00a0x<\/em>: [latex]\\hspace{.05in}-3|5x+2|\\le 12[\/latex]\r\n\r\n[reveal-answer q=\"647501\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"647501\"]\r\n\r\nFirst we must isolate the absolute value by dividing both sides by [latex]-3[\/latex].\u00a0 Remember that since we are dividing by a negative number, we must therefore reverse the inequality.\r\n

    [latex]\\displaystyle \\frac{|5x+2|}{-3}\\le \\frac{12}{-3}[\/latex]<\/p>\r\n

    [latex]\\displaystyle |5x+2|\\ge-4[\/latex]<\/p>\r\nThe absolute value of a quantity is always nonnegative, and therefore always greater than a negative number for all value of\u00a0x<\/em>.\u00a0 If follows that the solution set is all real numbers, [latex](-\\infty,\\infty)[\/latex].\r\n

    Answer<\/span><\/h4>\r\nAll Real Numbers\r\n\r\nInterval Notation: [latex](-\\infty,\\infty)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n \r\n

    Summary<\/h2>\r\nInequalities containing absolute value can be solved by rewriting them using compound inequalities. The first step to solving absolute inequalities is to isolate the absolute value. The next step is to decide whether you are working with an or<\/em> inequality or an and<\/em> inequality. If the inequality is greater than a number, we will use or<\/em>. If the inequality is less than a number, we will use and<\/em>. Remember that if we end up with an absolute value less than, or less than or equal to a negative number, there is no solution. If we end with an absolute value greater than a negative number, the solution is all real numbers.","rendered":"
    \n

    section 2.6 Learning Objectives<\/h3>\n

    2.6:\u00a0 Absolute Value Inequalities<\/strong><\/p>\n

      \n
    • Determine whether an absolute value inequality corresponds to a union or an intersection of inequalities<\/li>\n
    • Solve absolute value inequalities and express the solutions graphically and in interval notation<\/li>\n
    • Recognize when an absolute value inequality has no solution or all real numbers as the solution<\/li>\n<\/ul>\n<\/div>\n

       <\/p>\n

      Solve Inequalities Containing Absolute Value<\/h2>\n

      Let us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us start with a simple inequality.<\/p>\n

      [latex]\\left|x\\right|\\leq 4[\/latex]<\/p>\n

      This inequality is read, \u201cthe absolute value of x <\/i>is less than or equal to\u00a0[latex]4[\/latex].\u201d If you are asked to solve for x<\/i>, you want to find out what values of x <\/i>are\u00a0[latex]4[\/latex] units or less away from\u00a0[latex]0[\/latex] on a number line. You could start by thinking about the number line and what values of x <\/i>would satisfy this equation.<\/p>\n

      [latex]4[\/latex] and [latex]\u22124[\/latex] are both four units away from\u00a0[latex]0[\/latex], so they are solutions.\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are also solutions because each of these values is less than [latex]4[\/latex] units away from\u00a0[latex]0[\/latex]. So are\u00a0[latex]1[\/latex] and [latex]\u22121[\/latex],[latex]0.5[\/latex] and [latex]\u22120.5[\/latex], and so on\u2014there are an infinite number of values for x<\/i> that will satisfy this inequality.<\/p>\n

      The graph of this inequality will have two closed circles, at\u00a0[latex]4[\/latex] and [latex]\u22124[\/latex]. The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.<\/p>\n

      \"Number<\/p>\n

      The solution can be written this way:<\/p>\n

      Inequality notation: [latex]-4\\leq x\\leq4[\/latex]<\/p>\n

      Interval notation: [latex]\\left[-4,4\\right][\/latex]<\/p>\n

      The situation is a little different when the inequality sign is \u201cgreater than\u201d or \u201cgreater than or equal to.\u201d Consider the simple inequality [latex]\\left|x\\right|>3[\/latex]. Again, you could think of the number line and what values of x<\/i> are greater than\u00a0[latex]3[\/latex] units away from zero. This time,\u00a0[latex]3[\/latex] and [latex]\u22123[\/latex] are not included in the solution, so there are open circles on both of these values.\u00a0[latex]2[\/latex] and [latex]\u22122[\/latex] would not be solutions because they are not more than [latex]3[\/latex] units away from\u00a0[latex]0[\/latex]. But\u00a0[latex]5[\/latex] and [latex]\u22125[\/latex] would work and so would all of the values extending to the left of [latex]\u22123[\/latex] and to the right of\u00a0[latex]3[\/latex]. The graph would look like the one below.<\/p>\n

      \"Number<\/p>\n

      The solution to this inequality can be written this way:<\/p>\n

      Inequality notation:<\/i> [latex]x<\u22123[\/latex] or [latex]x>3[\/latex].<\/p>\n

      Interval notation: [latex]\\left(-\\infty, -3\\right)\\cup\\left(3,\\infty\\right)[\/latex]<\/p>\n

      In the following video, you will see examples of how to solve and express the solution\u00a0to absolute value inequalities involving both and<\/em> and or<\/em>.<\/p>\n