{"id":4103,"date":"2025-09-18T18:24:37","date_gmt":"2025-09-18T18:24:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=4103"},"modified":"2026-03-20T19:14:33","modified_gmt":"2026-03-20T19:14:33","slug":"systems-of-nonlinear-equations-and-inequalities-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/systems-of-nonlinear-equations-and-inequalities-learn-it-5\/","title":{"raw":"Systems of Nonlinear Equations and Inequalities: Learn It 5","rendered":"Systems of Nonlinear Equations and Inequalities: Learn It 5"},"content":{"raw":"

Graphing a System of Nonlinear Inequalities<\/h2>\r\nNow that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear.\r\n\r\nGraphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region<\/strong>.\r\n\r\n
How To: Given a system of nonlinear inequalities, sketch a graph.<\/strong>\r\n
    \r\n \t
  1. Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\r\n \t
  2. Graph the nonlinear equations.<\/li>\r\n \t
  3. Find the shaded regions of each inequality.<\/li>\r\n \t
  4. Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\r\n<\/ol>\r\n<\/section>
    Graph the given system of inequalities.\r\n

    [latex]\\begin{gathered} {x}^{2}-y\\le 0\\\\ 2{x}^{2}+y\\le 12\\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"801753\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"801753\"]\r\n\r\nThese two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both equations and the variable [latex]y[\/latex] will be eliminated. Then we solve for [latex]x[\/latex].\r\n

    [latex]\\begin{align} x^{2}\u2212y&=0 \\\\ 2x^{2}+y&=12 \\\\ \\hline 3x^{2}&=12 \\\\ x^{2}&=4 \\\\ x&=\\pm 2\\end{align}[\/latex]<\/p>\r\nSubstitute the [latex]x[\/latex]-values into one of the equations and solve for [latex]y[\/latex].\r\n

    [latex]\\begin{align} {x}^{2}-y&=0\\\\ {\\left(2\\right)}^{2}-y&=0\\\\ 4-y&=0\\\\ y&=4\\\\[5mm] {\\left(-2\\right)}^{2}-y&=0\\\\ 4-y&=0\\\\ y&=4\\end{align}[\/latex]<\/p>\r\nThe two points of intersection are [latex]\\left(2,4\\right)[\/latex] and [latex]\\left(-2,4\\right)[\/latex]. Notice that the equations can be rewritten as follows.\r\n

    [latex]\\begin{align}{x}^{2}-y&\\le 0 \\\\ {x}^{2}&\\le y \\\\ y&\\ge {x}^{2}\\end{align}[\/latex]<\/p>\r\n \r\n

    [latex]\\begin{gathered} 2{x}^{2}+y\\le 12 \\\\ y\\le -2{x}^{2}+12 \\end{gathered}[\/latex]<\/p>\r\nGraph each inequality.\u00a0The feasible region is the region between the two equations bounded by [latex]2{x}^{2}+y\\le 12[\/latex] on the top and [latex]{x}^{2}-y\\le 0[\/latex] on the bottom.\r\n\r\n\"Two\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    [ohm_question hide_question_numbers=1]321633[\/ohm_question]<\/section>
    \r\n
    \r\n\r\nGraph the given system of inequalities.\r\n

    [latex]\\begin{gathered}y\\ge {x}^{2}-1 \\\\ x-y\\ge -1 \\end{gathered}[\/latex]<\/p>\r\n[reveal-answer q=\"203187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"203187\"]\r\n\r\nShade the area bounded by the two curves, above the quadratic and below the line.\r\n\"A\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>","rendered":"

    Graphing a System of Nonlinear Inequalities<\/h2>\n

    Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities<\/strong> is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear.<\/p>\n

    Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region<\/strong>.<\/p>\n

    How To: Given a system of nonlinear inequalities, sketch a graph.<\/strong><\/p>\n
      \n
    1. Find the intersection points by solving the corresponding system of nonlinear equations.<\/li>\n
    2. Graph the nonlinear equations.<\/li>\n
    3. Find the shaded regions of each inequality.<\/li>\n
    4. Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points common to each inequality.<\/li>\n<\/ol>\n<\/section>\n
      Graph the given system of inequalities.<\/p>\n

      [latex]\\begin{gathered} {x}^{2}-y\\le 0\\\\ 2{x}^{2}+y\\le 12\\end{gathered}[\/latex]<\/p>\n