{"id":6721,"date":"2020-10-08T15:46:02","date_gmt":"2020-10-08T15:46:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6721"},"modified":"2026-01-13T07:04:37","modified_gmt":"2026-01-13T07:04:37","slug":"2-4-solving-compound-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/2-4-solving-compound-inequalities\/","title":{"raw":"2.4: Solving Compound Inequalities","rendered":"2.4: Solving Compound Inequalities"},"content":{"raw":"
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section 2.4 Learning Objectives<\/h3>\r\n2.4:\u00a0 Solving Compound Inequalities<\/strong>\r\n
    \r\n \t
  • Solve compound inequalities of the form of OR and express the solution graphically and in interval notation (union\/disjunction)<\/li>\r\n \t
  • Solve compound inequalities of the form AND and express the solution graphically and in interval notation (intersection\/conjunction)<\/li>\r\n \t
  • Solve tripartite inequalities and express the solution graphically and in interval notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nIn this section you will learn to:<\/strong>\r\n
      \r\n \t
    • Solve compound inequalities\u2014OR\r\n
        \r\n \t
      • Solve compound inequalities in the form of or<\/i> and express the solution graphically and with an interval<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • Solve compound inequalities\u2014AND\r\n
          \r\n \t
        • Express solutions to inequalities graphically and with interval notation<\/li>\r\n \t
        • Identify solutions for compound inequalities in the form [latex]a<x<b[\/latex], including cases with no solution<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n

          Disjunctions: Solve compound inequalities in the form of or <\/i>and express the solution graphically and in interval notation<\/h2>\r\nAs we saw in the last section, the solution of a compound inequality that consists of two inequalities joined with the word or<\/em> is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.\r\n\r\nIn this section you will see that some inequalities need to be simplified before their solution can be written or graphed.\r\n\r\nIn the following example, you will see an example of how to solve a one-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.\r\n
          \r\n

          Example 1<\/h3>\r\nSolve for x:<\/em>\u00a0 [latex]\\hspace{.05in} x\u20135>0[\/latex] or\u00a0[latex]3x\u20131<8[\/latex]\r\n\r\n[reveal-answer q=\"212910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"212910\"]Solve each inequality by isolating the variable.\r\n

          [latex] \\displaystyle \\begin{array}{r}x-5>0\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1<8\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,+1}\\\\x\\,\\,\\,\\,\\,\\,>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,<\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x<3\\,\\,\\,\\\\x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nWrite both inequality solutions as a compound inequality using or, <\/i>and\u00a0using interval notation.\r\n

          Answer<\/h4>\r\nInequality: [latex] \\displaystyle x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]\r\n\r\nThe solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first before writing the solution using interval notation.\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRemember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.\r\n
          \r\n

          Example 2<\/h3>\r\nSolve for y:<\/em>\u00a0 [latex]\\hspace{.05in} 2y+7\\lt13\\text{ or }\u22123y\u20132\\le10[\/latex]\r\n[reveal-answer q=\"969462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"969462\"]\r\n\r\nSolve each inequality separately.\r\n

          [latex] \\displaystyle \\begin{array}{l}2y+7<13\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-3y-2\\le 10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,\\,\\,+2}\\\\\\frac{2y}{2}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\frac{6}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-3y}{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{12}{-3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\ge -4\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,y\\ge -4\\end{array}[\/latex]<\/p>\r\nThe inequality sign is reversed with division by a negative number.\r\n\r\nSince y<\/i> could be less than 3 or greater than or equal to [latex]\u22124[\/latex], y<\/i> could be any number. Graphing the inequality helps with this interpretation.\r\n

          Answer<\/h4>\r\nInequality: [latex]y<3\\text{ or }y\\ge -4[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,\\infty\\right)[\/latex]\r\n\r\nGraph:\r\n\r\n\"Number\r\n\r\nSince every<\/em> number is part of the solution set, an appropriate final graph is shown below:\r\n\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution \"all real numbers.\" \u00a0Any real number will produce a true statement for either\u00a0[latex]y<3\\text{ or }y\\ge -4[\/latex] when it is substituted for x<\/em>.\r\n
          \r\n

          Example 3<\/h3>\r\nSolve for z:<\/em>\u00a0[latex]\\hspace{.05in} 5z\u20133\\gt\u221218[\/latex] or [latex]\u22122z\u20131\\gt15[\/latex]\r\n[reveal-answer q=\"74043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"74043\"]\r\n\r\nSolve each inequality separately.\u00a0Combine the solutions.\r\n

          [latex] \\displaystyle \\begin{array}{l}5z-3>-18\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-2z-1>15\\\\\\underline{\\,\\,\\,\\,\\,\\,+3\\,\\,\\,\\,\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,+1}\\\\\\frac{5z}{5}\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\frac{-15}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-2z}{-2}\\,\\,\\,\\,\\,\\,>\\,\\,\\frac{16}{-2}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z<-8\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality:\u00a0[latex] \\displaystyle z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[\/latex] has solutions that continue all the way to the right. In this way we write solutions with intervals from left to right.\r\n\r\nGraph:\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video contains an example of solving a compound inequality involving OR, and drawing the associated graph.\r\n\r\nhttps:\/\/youtu.be\/oRlJ8G7trR8\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Possible Cases for Disjunction<\/strong><\/td>\r\n<\/tr>\r\n
          Case 1:<\/td>\r\n<\/tr>\r\n
          Description<\/td>\r\nThe solution could be the union of disjoint sets extending in opposite directions.<\/td>\r\n<\/tr>\r\n
          Example of\u00a0 Inequalities<\/td>\r\n[latex]x\\le{-1}[\/latex] or [latex]x\\gt{1}[\/latex]<\/td>\r\n<\/tr>\r\n
          Initial Intervals<\/td>\r\n[latex](-\\infty,-1] \\mbox{ or } (1,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n
          Graphs<\/td>\r\n\r\n

          \"Number<\/p>\r\nSince we want the union of the two regions, both shaded regions will be included in our final graph.\r\n

          \"Number<\/p>\r\n<\/td>\r\n<\/tr>\r\n

          Final Answer in Interval Notation<\/td>\r\n\r\n

          [latex](-\\infty,-1] \\cup (1,\\infty)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

          Case 2:<\/td>\r\n<\/tr>\r\n
          Description<\/td>\r\nThe solution could begin at a point on the number line and extend in one direction.<\/td>\r\n<\/tr>\r\n
          Example of\u00a0 Inequalities<\/td>\r\n[latex]x\\gt -3[\/latex] or [latex]x\\ge4[\/latex]<\/td>\r\n<\/tr>\r\n
          Initial Intervals<\/td>\r\n[latex]\\left(-3,\\infty\\right) \\mbox{ or } [4,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n
          Graphs<\/td>\r\n\r\n

          \u00a0\"Number<\/p>\r\nSince the union includes both regions, everything to the right of [latex]-3[\/latex] will be shaded in the final graph.\r\n

          \"Number<\/p>\r\n<\/td>\r\n<\/tr>\r\n

          Final Answer in Interval Notation<\/td>\r\n\r\n

          [latex](-3,\\infty)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

          Case 3:<\/td>\r\n<\/tr>\r\n
          \u00a0Description<\/td>\r\nThe solution could be all real numbers, covering the entire number line.<\/td>\r\n<\/tr>\r\n
          \u00a0Example of\u00a0 \u00a0Inequalities<\/td>\r\n[latex]x\\gt{-3}[\/latex] or [latex]x\\lt{3}[\/latex]<\/td>\r\n<\/tr>\r\n
          \u00a0Initial Intervals<\/td>\r\n[latex]\\left(-3,\\infty\\right) \\mbox{ or } \\left(-\\infty,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          \u00a0Graph<\/td>\r\n\r\n

          \"Number<\/p>\r\nEvery value is included in one (or both) of the shaded regions.\u00a0 So, the final graph will have the entire number line shaded.\r\n

          \"Number<\/p>\r\n<\/td>\r\n<\/tr>\r\n

          Final Answer in Interval Notation<\/td>\r\n\r\n

          [latex](-\\infty,\\infty)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the next section you will see examples of how to solve compound inequalities containing and<\/em>.\r\n

          Conjunctions: Solve compound inequalities in the form of and<\/i> and express the solution graphically and in interval notation<\/h2>\r\nThe solution of a compound inequality that consists of two inequalities joined with the word and <\/i>is the intersection of the solutions of each inequality. In other words, both statements must be true at the same time. The solution to an and<\/i> compound inequality are all the solutions that the two inequalities have in common. As we saw in the last sections, this is\u00a0where the two graphs overlap.\r\n\r\nIn this section we will see more examples where we have to simplify the compound inequalities before we can express their solutions graphically or with an interval.\r\n
          \r\n

          Example 4<\/h3>\r\nSolve for x:<\/i>\u00a0[latex]\\hspace{.05in} \\displaystyle 1-4x\\le 21\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,5x+2\\ge22[\/latex]\r\n\r\n[reveal-answer q=\"266032\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"266032\"]\r\n\r\nSolve each inequality for x<\/em>.\u00a0Determine the intersection of the solutions.\r\n

          [latex] \\displaystyle \\begin{array}{r}\\,\\,\\,1-4x\\le 21\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,5x+2\\ge 22\\\\\\underline{-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-1}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-2\\,\\,\\,\\,-2}\\\\\\,\\,\\,\\,\\,\\underline{-4x}\\leq \\underline{20}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{5x}\\,\\,\\,\\,\\,\\,\\,\\ge \\underline{20}\\\\\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,{-4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{5}\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge -5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 4\\,\\,\\,\\,\\\\\\\\x\\ge -5\\,\\text{and}\\,\\,x\\ge 4\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThe number line below shows the graphs of the two inequalities in the problem. The solution to the compound inequality is [latex]x\\geq4[\/latex], since\u00a0this is where the two graphs overlap.\r\n\r\n\"Number\r\n

          Answer<\/h4>\r\nInequality: [latex] \\displaystyle x\\ge 4[\/latex]\r\n\r\nInterval: [latex]\\left[4,\\infty\\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

          Example 5<\/h3>\r\nSolve for x<\/em>: \u00a0[latex]\\hspace{.05in} \\displaystyle {5}{x}-{2}\\le{3}\\text{ and }{4}{x}{+7}>{3}[\/latex]\r\n[reveal-answer q=\"784358\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"784358\"]\r\n\r\nSolve each inequality separately.\u00a0Find the overlap between the solutions.\r\n

          [latex] \\displaystyle \\begin{array}{l}\\,\\,\\,5x-2\\le 3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,4x+7>\\,\\,\\,\\,3\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,+2\\,\\,+2\\,}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,\\,\\,-7}\\\\\\,\\,\\frac{5x}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{5}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{4x}{4}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\frac{-4}{4}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>-1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\le 1\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x>-1\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex]-1\\lt{x}\\le{1}[\/latex]\r\n\r\nInterval: [latex]\\left(-1,1\\right][\/latex]\r\n\r\nGraph:\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          Tripartite Inequalities: Compound inequalities in the form [latex]a<x<b[\/latex]<\/h2>\r\nCompound inequalities in the form [latex]a<x<b[\/latex] are sometimes called \"tripartite inequalities,\" and correspond to compound inequalities adjoined by\u00a0and<\/em>.\u00a0 With these problems, you can split a compound inequality in the form of\u00a0\u00a0[latex]a<x<b[\/latex]\u00a0into two inequalities [latex]x<b[\/latex] and <\/i>[latex]x>a[\/latex], or you can solve the inequality more quickly by applying the properties of inequality to all three segments of the compound inequality.\r\n\r\nIn the example below, we will show how to apply the properties of inequality to all three segments of the compound inequality. In the video below the example, we will show how to split it into two inequalities to solve. Read and view both examples to see if you prefer one method over the other.\r\n
          \r\n

          Example 6<\/h3>\r\nSolve for x:<\/i>\u00a0[latex]\\hspace{.05in} 3\\lt2x+3\\leq 7[\/latex]\r\n\r\n[reveal-answer q=\"39150\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"39150\"]\r\n\r\n \r\n\r\nIsolate the variable by subtracting 3 from all 3 parts of the inequality, then dividing each part by 2.\r\n

          [latex]\\begin{array}{r}\\,\\,\\,\\,3\\,\\,\\lt\\,\\,2x+3\\,\\,\\leq \\,\\,\\,\\,7\\\\\\underline{\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,-3}\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,-3}\\,\\\\\\,\\,\\,\\,\\,\\underline{\\,0\\,}\\,\\,\\lt\\,\\,\\,\\,\\underline{2x}\\,\\,\\,\\,\\,\\,\\,\\,\\leq\\,\\,\\,\\underline{\\,4\\,}\\\\2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,0\\lt x\\leq 2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n

          Answer<\/h4>\r\nInequality: [latex] \\displaystyle 0\\lt{x}\\le 2[\/latex]\r\n\r\nInterval: [latex]\\left(0,2\\right][\/latex]\r\n\r\nGraph:\r\n\r\n\"Number\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

          <\/h2>\r\nThe following video works through two examples of solving tripartite inequalities.\r\n\r\nhttps:\/\/youtu.be\/mV7xjGipcYo\r\n\r\n \r\n\r\nAlternately, a tripartite inequality can be solved by formally breaking it into an\u00a0and<\/em> compound inequality, as shown in the video below.\r\n\r\nhttps:\/\/youtu.be\/UU_KJI59_08\r\n\r\nTo solve inequalities of the form [latex]a<x<b[\/latex], either rewrite as an and<\/em> compound inequality or\u00a0use the addition and multiplication properties of inequality to solve the tripartite inequality for x <\/i>directly. Whatever operation you perform on the middle portion of the inequality, you must also perform to each of the outside sections as well. Pay particular attention to division or multiplication by a negative.\r\n\r\nThe solution to a compound inequality with and<\/i> is always the overlap between the solution to each inequality. There are three possible outcomes for compound inequalities joined by the word and<\/i>:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Possible Cases for Conjunction<\/strong><\/td>\r\n<\/tr>\r\n
          Case 1:<\/td>\r\n<\/tr>\r\n
          Description<\/td>\r\nThe solution could be all the values between two endpoints<\/td>\r\n<\/tr>\r\n
          Example of Inequalities<\/td>\r\n[latex]x\\le{1}[\/latex] and [latex]x\\gt{-1}[\/latex], or as a bounded inequality: [latex]{-1}\\lt{x}\\le{1}[\/latex]<\/td>\r\n<\/tr>\r\n
          Initial Intervals<\/td>\r\n[latex]\\left(-\\infty,1\\right][\/latex] and [latex]\\left(-1,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          Graphs<\/td>\r\n\"Number\r\n\r\nThe intersection is the overlap between the two regions, which will include only values between [latex]-1[\/latex] and [latex]1[\/latex] (not including [latex]-1[\/latex] itself).\r\n\r\n\"Number<\/td>\r\n<\/tr>\r\n
          Final Answer in Interval Notation<\/td>\r\n[latex](-1,1][\/latex]<\/td>\r\n<\/tr>\r\n
          Case 2:<\/td>\r\n<\/tr>\r\n
          Description<\/td>\r\nThe solution could begin at a point on the number line and extend in one direction.<\/td>\r\n<\/tr>\r\n
          Example of Inequalities<\/td>\r\n[latex]x\\gt -3[\/latex] and [latex]x\\ge4[\/latex]<\/td>\r\n<\/tr>\r\n
          Initial Intervals<\/td>\r\n[latex](-3,\\infty)[\/latex] and\u00a0 [latex]\\left[4,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          Graphs<\/td>\r\n\u00a0\"Number\r\n\r\nThe two shaded regions overlap for all values greater than or equal to [latex]4[\/latex].\r\n\r\n\"Number<\/td>\r\n<\/tr>\r\n
          Final Answer in Interval Notation<\/td>\r\n[latex][4,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n
          Case 3:<\/td>\r\n<\/tr>\r\n
          \u00a0Description<\/td>\r\nThere is no solution to the compound inequality<\/td>\r\n<\/tr>\r\n
          Example of Inequalities<\/td>\r\n[latex]x\\lt{-3}[\/latex] and [latex]x\\gt{3}[\/latex]<\/td>\r\n<\/tr>\r\n
          \u00a0Intervals<\/td>\r\n[latex]\\left(-\\infty,-3\\right)[\/latex] and [latex]\\left(3,\\infty\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
          \u00a0Graph<\/td>\r\n\"Number\r\n\r\nSince there is no overlap, there is no solution to highlight on the graph as shown below.\r\n\r\n\"A<\/td>\r\n<\/tr>\r\n
          Final Answer in Interval Notation<\/td>\r\nNo Solution (equivalently, the empty set [latex]\\{ \\hspace{.05in} \\}[\/latex] or [latex]\\varnothing[\/latex])<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the example below, there is no solution to the compound inequality because there is no overlap between the inequalities.\r\n
          \r\n

          Example 7<\/h3>\r\nSolve for x:<\/em>\u00a0[latex]\\hspace{.05in} x+2>5[\/latex] and [latex]x+4<5[\/latex]\r\n\r\n[reveal-answer q=\"336256\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"336256\"]\r\n\r\nSolve each inequality separately.\r\n

          [latex] \\displaystyle \\begin{array}{l}x+2>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{AND}\\,\\,\\,\\,\\,\\,\\,x+4<5\\,\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,-2\\,-2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,-4\\,-4}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\,\\,\\,\\,<\\,1\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x>3\\,\\,\\,\\,\\text{and}\\,\\,\\,\\,x<1\\end{array}[\/latex]<\/p>\r\nFind the overlap between the solutions.\r\n\r\n\"Number\r\n

          Answer<\/h4>\r\nThere is no overlap between [latex] \\displaystyle x>3[\/latex] and [latex]x<1[\/latex], so there is no solution. Since there is no overlap, there is no solution to highlight on the graph as shown below.\r\n\r\n \r\n\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"
          \n

          section 2.4 Learning Objectives<\/h3>\n

          2.4:\u00a0 Solving Compound Inequalities<\/strong><\/p>\n

            \n
          • Solve compound inequalities of the form of OR and express the solution graphically and in interval notation (union\/disjunction)<\/li>\n
          • Solve compound inequalities of the form AND and express the solution graphically and in interval notation (intersection\/conjunction)<\/li>\n
          • Solve tripartite inequalities and express the solution graphically and in interval notation<\/li>\n<\/ul>\n<\/div>\n

             <\/p>\n

            In this section you will learn to:<\/strong><\/p>\n

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            • Solve compound inequalities\u2014OR\n
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              • Solve compound inequalities in the form of or<\/i> and express the solution graphically and with an interval<\/li>\n<\/ul>\n<\/li>\n
              • Solve compound inequalities\u2014AND\n
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                • Express solutions to inequalities graphically and with interval notation<\/li>\n
                • Identify solutions for compound inequalities in the form [latex]a\n<\/ul>\n<\/li>\n<\/ul>\n

                  Disjunctions: Solve compound inequalities in the form of or <\/i>and express the solution graphically and in interval notation<\/h2>\n

                  As we saw in the last section, the solution of a compound inequality that consists of two inequalities joined with the word or<\/em> is the union of the solutions of each inequality. Unions allow us to create a new set from two that may or may not have elements in common.<\/p>\n

                  In this section you will see that some inequalities need to be simplified before their solution can be written or graphed.<\/p>\n

                  In the following example, you will see an example of how to solve a one-step inequality in the OR form. Note how each inequality is treated independently until the end where the solution is described in terms of both inequalities. You will use the same properties to solve compound inequalities that you used to solve regular inequalities.<\/p>\n

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                  Example 1<\/h3>\n

                  Solve for x:<\/em>\u00a0 [latex]\\hspace{.05in} x\u20135>0[\/latex] or\u00a0[latex]3x\u20131<8[\/latex]\n\n\n\n

                  Show Solution<\/span><\/p>\n
                  Solve each inequality by isolating the variable.<\/p>\n

                  [latex]\\displaystyle \\begin{array}{r}x-5>0\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3x-1<8\\\\\\underline{\\,\\,\\,+5\\,\\,+5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+1\\,+1}\\\\x\\,\\,\\,\\,\\,\\,>5\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{3x}\\,\\,\\,<\\underline{9}\\\\{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{3}\\\\x<3\\,\\,\\,\\\\x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n

                  Write both inequality solutions as a compound inequality using or, <\/i>and\u00a0using interval notation.<\/p>\n

                  Answer<\/h4>\n

                  Inequality: [latex]\\displaystyle x>5\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,x<3[\/latex]\n\nInterval: [latex]\\left(-\\infty, 3\\right)\\cup\\left(5,\\infty\\right)[\/latex]\n\nThe solution to this compound inequality can also be shown graphically. Sometimes it helps to draw the graph first before writing the solution using interval notation.\n\n\"Number<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                  Remember to apply the properties of inequality when you are solving compound inequalities. The next example involves dividing by a negative to isolate a variable.<\/p>\n

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                  Example 2<\/h3>\n

                  Solve for y:<\/em>\u00a0 [latex]\\hspace{.05in} 2y+7\\lt13\\text{ or }\u22123y\u20132\\le10[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
                  \n

                  Solve each inequality separately.<\/p>\n

                  [latex]\\displaystyle \\begin{array}{l}2y+7<13\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-3y-2\\le 10\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,-7\\,\\,\\,\\,-7}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,+2\\,\\,\\,\\,\\,+2}\\\\\\frac{2y}{2}\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\frac{6}{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-3y}{-3}\\,\\,\\,\\,\\,\\,\\,\\,\\le \\frac{12}{-3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y\\ge -4\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,y<3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,y\\ge -4\\end{array}[\/latex]<\/p>\n

                  The inequality sign is reversed with division by a negative number.<\/p>\n

                  Since y<\/i> could be less than 3 or greater than or equal to [latex]\u22124[\/latex], y<\/i> could be any number. Graphing the inequality helps with this interpretation.<\/p>\n

                  Answer<\/h4>\n

                  Inequality: [latex]y<3\\text{ or }y\\ge -4[\/latex]\n\nInterval: [latex]\\left(-\\infty,\\infty\\right)[\/latex]\n\nGraph:\n\n\"Number<\/p>\n

                  Since every<\/em> number is part of the solution set, an appropriate final graph is shown below:<\/p>\n

                  \"A<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                  In the last example, the final answer included solutions whose intervals overlapped, causing the answer to include all the numbers on the number line. In words, we call this solution “all real numbers.” \u00a0Any real number will produce a true statement for either\u00a0[latex]y<3\\text{ or }y\\ge -4[\/latex] when it is substituted for x<\/em>.<\/p>\n

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                  Example 3<\/h3>\n

                  Solve for z:<\/em>\u00a0[latex]\\hspace{.05in} 5z\u20133\\gt\u221218[\/latex] or [latex]\u22122z\u20131\\gt15[\/latex]<\/p>\n

                  Show Solution<\/span><\/p>\n
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                  Solve each inequality separately.\u00a0Combine the solutions.<\/p>\n

                  [latex]\\displaystyle \\begin{array}{l}5z-3>-18\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{OR}\\,\\,\\,\\,\\,\\,\\,-2z-1>15\\\\\\underline{\\,\\,\\,\\,\\,\\,+3\\,\\,\\,\\,\\,\\,\\,+3}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,+1\\,\\,\\,\\,+1}\\\\\\frac{5z}{5}\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\frac{-15}{5}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\frac{-2z}{-2}\\,\\,\\,\\,\\,\\,>\\,\\,\\frac{16}{-2}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z<-8\\\\\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8\\end{array}[\/latex]<\/p>\n

                  Answer<\/h4>\n

                  Inequality:\u00a0[latex]\\displaystyle z>-3\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,z<-8[\/latex]\n\nInterval: [latex]\\left(-\\infty,-8\\right)\\cup\\left(-3,\\infty\\right)[\/latex] Note how we write the intervals with the one containing the most negative solutions first, then move to the right on the number line. [latex]z<-8[\/latex] has solutions that continue all the way to the left on the number line, whereas [latex]x>-3[\/latex] has solutions that continue all the way to the right. In this way we write solutions with intervals from left to right.<\/p>\n

                  Graph:\"Number<\/p>\n<\/div>\n<\/div>\n<\/div>\n

                  The following video contains an example of solving a compound inequality involving OR, and drawing the associated graph.<\/p>\n