{"id":1043,"date":"2025-07-21T23:26:15","date_gmt":"2025-07-21T23:26:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1043"},"modified":"2026-01-14T19:24:56","modified_gmt":"2026-01-14T19:24:56","slug":"applications-of-rational-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/applications-of-rational-functions-learn-it-3\/","title":{"raw":"Applications of Rational Functions: Learn It 3","rendered":"Applications of Rational Functions: Learn It 3"},"content":{"raw":"

Solving a Rational Inequality<\/h2>\r\n

A rational inequality is similar to a rational equation\u2014it contains one or more rational expressions\u2014but instead of an equal sign, it includes an inequality symbol like [latex]<[\/latex], [latex]\\leq[\/latex], [latex]>[\/latex], or [latex]\\geq[\/latex].<\/p>\r\n\r\n

\r\n

rational inequality<\/h3>\r\nA rational inequality is an inequality that contains at least one rational expression where the variable appears in a denominator.\r\n\r\n<\/section>
Solve the rational inequality:[latex]\\dfrac{x - 4}{x + 2} > 0[\/latex][reveal-answer q=\"56901\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"56901\"]Step 1: Identify critical values The numerator is zero when [latex]x = 4[\/latex] The denominator is zero when [latex]x = -2[\/latex] (excluded from the solution) These values divide the number line into intervals:\r\n
    \r\n \t
  • Interval 1: [latex]x < -2[\/latex]<\/li>\r\n \t
  • Interval 2: [latex]-2 < x < 4[\/latex]<\/li>\r\n \t
  • Interval 3: [latex]x > 4[\/latex]<\/li>\r\n<\/ul>\r\nStep 2: Test a point from each interval\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Interval<\/td>\r\nTest Point<\/td>\r\nSign of [latex]\\frac{x - 4}{x + 2}[\/latex]<\/td>\r\n<\/tr>\r\n
    \u00a0[latex]x < -2[\/latex]<\/td>\r\n\u00a0[latex]-3[\/latex]<\/td>\r\n[latex]\\frac{-7}{-1} = 7[\/latex] \u2192 positive<\/td>\r\n<\/tr>\r\n
    [latex]-2 < x < 4[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]\\frac{-4}{2} = -2[\/latex] \u2192 negative<\/td>\r\n<\/tr>\r\n
    [latex]x > 4[\/latex]<\/td>\r\n\u00a0[latex]5[\/latex]<\/td>\r\n[latex]\\frac{1}{7}[\/latex] \u2192 positive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe are looking for where the expression is greater than 0, so we want the intervals where the expression is positive.\r\n\r\nStep 3: Write the solution The solution is: [latex](-\\infty, -2) \\cup (4, \\infty)[\/latex]\r\n\r\nNote: We use open intervals because the expression is strictly greater than 0 [latex]x = -2[\/latex] makes the expression undefined [latex]x = 4[\/latex] makes the numerator zero, which gives 0\u2014not greater than 0[\/hidden-answer]\r\n\r\n<\/section>
    To solve a rational inequality:<\/strong>\r\n
      \r\n \t
    1. Write the inequality so that one side is zero and the other side is a single rational expression<\/li>\r\n \t
    2. Identify critical values (zeros of the numerator and undefined points from the denominator)<\/li>\r\n \t
    3. Use a sign chart to test the sign of the expression in each interval<\/li>\r\n \t
    4. Determine the interval(s) where the expression satisfies the inequality<\/li>\r\n<\/ol>\r\n<\/section>
      \r\n

      Common Mistakes When Solving Rational Inequalities<\/strong><\/p>\r\n\r\n

        \r\n \t
      • \r\n

        Including excluded values in the solution:\r\nAlways identify and exclude values that make the denominator zero.<\/p>\r\n<\/li>\r\n \t

      • \r\n

        Not setting one side to zero before testing signs:\r\nThe sign chart method only works when the inequality is compared to zero.<\/p>\r\n<\/li>\r\n \t

      • \r\n

        Forgetting open vs. closed intervals:\r\nIf the inequality is strict ([latex]<[\/latex] or [latex]>[\/latex]), use open intervals for all values. If it includes equality ([latex]\\leq[\/latex] or [latex]\\geq[\/latex]), to include values that make the numerator zero (but still exclude points that make the denominator zero).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/section>

        \r\n

        Solve the rational inequality:<\/p>\r\n

        [latex]\\dfrac{x + 3}{x - 5} \\leq 0[\/latex]<\/p>\r\n[reveal-answer q=\"218664\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"218664\"]\r\n\r\nStep 1: Identify critical values\r\n\r\nNumerator: [latex]x + 3 = 0 \\Rightarrow x = -3[\/latex]\r\n\r\nDenominator: [latex]x - 5 = 0 \\Rightarrow x = 5[\/latex] (excluded)\r\n\r\nIntervals to test:\r\n

          \r\n \t
        • [latex](-\\infty, -3)[\/latex]<\/li>\r\n \t
        • [latex](-3, 5)[\/latex]<\/li>\r\n \t
        • [latex](5, \\infty)[\/latex]<\/li>\r\n<\/ul>\r\nStep 2: Sign test\r\n\r\n\r\n\r\n\r\n\r\n\r\n
          Interval<\/td>\r\nTest Point<\/td>\r\nSign of [latex]\\frac{x +3}{x -5}[\/latex]<\/td>\r\n<\/tr>\r\n
          \u00a0[latex]x < -3[\/latex]<\/td>\r\n\u00a0[latex]-4[\/latex]<\/td>\r\n[latex]\\frac{-1}{-9}[\/latex] \u2192 positive<\/td>\r\n<\/tr>\r\n
          [latex]-3 < x < 5*[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]\\frac{3}{-5} [\/latex] \u2192 negative<\/td>\r\n<\/tr>\r\n
          [latex]x > 5*[\/latex]<\/td>\r\n\u00a0[latex]6[\/latex]<\/td>\r\n[latex]\\frac{9}{1}[\/latex] \u2192 positive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n*This is an asymptote, the rational function is undefined so we won't use [ ] for this value.\r\n\r\nWe are looking for [latex]\\leq 0[\/latex] so we want the intervals where the expression is negative.\r\n\r\nStep 3: Solution [latex][-3, 5)[\/latex][\/hidden-answer]\r\n\r\n<\/section>
          [ohm_question hide_question_numbers=1]317827[\/ohm_question]<\/section>
          [ohm_question hide_question_numbers=1]317828[\/ohm_question]<\/section><\/section>
          <\/section>","rendered":"

          Solving a Rational Inequality<\/h2>\n

          A rational inequality is similar to a rational equation\u2014it contains one or more rational expressions\u2014but instead of an equal sign, it includes an inequality symbol like [latex]<[\/latex], [latex]\\leq[\/latex], [latex]>[\/latex], or [latex]\\geq[\/latex].<\/p>\n

          \n

          rational inequality<\/h3>\n

          A rational inequality is an inequality that contains at least one rational expression where the variable appears in a denominator.<\/p>\n<\/section>\n

          Solve the rational inequality:[latex]\\dfrac{x - 4}{x + 2} > 0[\/latex]<\/p>\n