{"id":1145,"date":"2023-07-22T07:40:23","date_gmt":"2023-07-22T07:40:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/?post_type=chapter&#038;p=1145"},"modified":"2023-09-07T16:47:09","modified_gmt":"2023-09-07T16:47:09","slug":"solving-polynomial-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/solving-polynomial-inequalities\/","title":{"raw":"\u25aa   Solving Polynomial Inequalities","rendered":"\u25aa   Solving Polynomial Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Solve polynomial inequalities using boundary value method.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Solving Polynomial Inequalities using Boundary Value Method<\/h2>\r\nAny inequality that can be put into one of the following forms\r\n<p style=\"padding-left: 30px;\">[latex]f(x)&gt;0, f(x) \\geq 0, f(x)&lt;0,[\/latex] or [latex]f(x) \\leq 0[\/latex], where [latex]f[\/latex] is a polynomial function<\/p>\r\nis called <strong>polynomial inequality<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>How To: Solve Polynomial Inequalities using Boundary Value MEthod<\/h3>\r\n<ol>\r\n \t<li>Rewrite the given polynomial inequality as an equation by replacing the inequality symbol with the equal sign.<\/li>\r\n \t<li>Solve the polynomial equation. The real solution(s) of the equation is(are) the <strong>boundary point(s)<\/strong>.<\/li>\r\n \t<li>Plot the boundary point(s) from Step 2 on a number line.\r\n[latex] \\Rightarrow [\/latex] Use an open circle when the given inequality has [latex]&lt;[\/latex] or [latex]&gt;[\/latex]\r\n[latex]\\Rightarrow [\/latex] Use a closed circle when the given inequality has [latex]\\leq[\/latex] or [latex]\\geq[\/latex].<\/li>\r\n \t<li>Choose one number, which is called a <strong>test value<\/strong>, from each interval and test the intervals by evaluating the given inequality at that number.\r\n[latex]\\Rightarrow [\/latex] If the inequality is TRUE, then the interval is a solution of the inequality.\r\n[latex]\\Rightarrow[\/latex] If the inequality is FALSE, then the interval is not a solution of the inequality.<\/li>\r\n \t<li>Write the solution set (usually in interval notation), selecting the interval(s) from Step 4.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving Polynomial Inequality using Boundary Value Method<\/h3>\r\nSolve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.\r\n<p style=\"padding-left: 30px;\">[latex]x^4-4x^3+3x^2 &gt; 0[\/latex]<\/p>\r\n[reveal-answer q=\"933923\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"933923\"]\r\n\r\nFirst, let's rewrite the inequality as an equation and then solve it:\r\n<p style=\"padding-left: 30px;\">[latex]x^4-4x^3+3x^2=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2(x^2-4x+3)=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2(x-1)(x-3)=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2=0, x-1=0[\/latex] or [latex]x-3=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x=0, x=1[\/latex] or [latex]x=3[\/latex]<\/p>\r\nSecond, plot the solutions on the number line. Since the inequality has \"[latex]&gt;[\/latex]\" (greater than), which doesn't have \"[latex]=[\/latex]\" (equal sign), they should be <strong>open<\/strong> circles:\r\n<p style=\"padding-left: 30px;\"><img class=\"aligncenter wp-image-1154\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1.png\" alt=\"Numberline plotted 0, 1, 3 (open) as solutions, marked -1, 0.5, 2, 5 as testing values\" width=\"453\" height=\"74\" \/><\/p>\r\n<p style=\"padding-left: 30px;\">Note that now there are four intervals on the number line: [latex](-\\infty, 0), (0, 1), (1, 3),[\/latex] and [latex](3, \\infty)[\/latex]<\/p>\r\nThird, choose your choice of test value from each interval:\r\n<p style=\"padding-left: 30px;\">I chose [latex]-1, 0.5, 2,[\/latex] and [latex]5[\/latex] from those four intervals.<\/p>\r\nFourth, test each interval using the test value:\r\n<p style=\"padding-left: 30px;\">When [latex]x=-1[\/latex], [latex](-1)^4-4(-1)^3+3(-1)^2&gt;0 \\rightarrow 8&gt;0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=0.5[\/latex], [latex](0.5)^4-4(0.5)^3+3(0.5)^2&gt;0 \\rightarrow 0.3125&gt;0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=2[\/latex], [latex](2)^4-4(2)^3+3(2)^2&gt;0 \\rightarrow -4&gt;0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=5[\/latex], [latex](5)^4-4(5)^3+3(5)^2&gt;0 \\rightarrow 200&gt;0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<img class=\"aligncenter wp-image-1157\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2.png\" alt=\"Number line plotted 0, 1, 3 (open) as solutions, marked -1, 0.5, 2, 5 as testing values, first and fourth intervals are marked in red\" width=\"457\" height=\"71\" \/>\r\n<p style=\"padding-left: 30px;\">Actually, we can check the results of test values from the graph of [latex]f(x)=x^4-4x^3+3x^2[\/latex] below:<\/p>\r\n<img class=\"aligncenter wp-image-1158\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3.png\" alt=\"Graph of f(x)=x^4-4x^3+3x^2, marked above x-axis in red\" width=\"237\" height=\"284\" \/>\r\n<p style=\"padding-left: 30px;\">* As we can see from the graph, it doesn't matter which number you choose from each interval for testing. For example, any number that is greater than 5 will make the function value positive.<\/p>\r\nTherefore, the solution of the polynomial inequality\u00a0[latex]x^4-4x^3+3x^2 &gt; 0[\/latex] is [latex](-\\infty, 0) \\cup (0, 1) \\cup (3, \\infty)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.\r\n<p style=\"padding-left: 30px;\">[latex]x^2 \\leq 3x+4[\/latex]<\/p>\r\n[reveal-answer q=\"651999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"651999\"]\r\n\r\nFirst, let's rewrite the inequality as an equation and then solve it:\r\n<p style=\"padding-left: 30px;\">[latex]x^2 = 3x+4[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2-3x-4=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex](x+1)(x-4)=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x+1=0[\/latex] or [latex]x-4=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x=-1[\/latex] or [latex]x=4[\/latex]<\/p>\r\nSecond, plot the solutions on the number line. Since the inequality has \"[latex]\\leq[\/latex]\" (less than or equal to), which has \"[latex]=[\/latex]\" (equal sign), they should be <strong>closed<\/strong> circles:\r\n\r\n<img class=\"aligncenter wp-image-1159\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2.png\" alt=\"Number line plotted -1, 4 (closed) as solutions, marked -2, 0, 5 as test values\" width=\"392\" height=\"76\" \/>\r\n<p style=\"padding-left: 30px;\">Note that now there are three intervals on the number line: [latex](-\\infty, -1], [-1, 4],[\/latex] and [latex][4, \\infty)[\/latex]<\/p>\r\nThird, choose your choice of test value from each interval:\r\n<p style=\"padding-left: 30px;\">I chose [latex]-2, 0,[\/latex] and [latex]5[\/latex] from those three intervals.<\/p>\r\nFourth, test each interval using the test value:\r\n<p style=\"padding-left: 30px;\">When [latex]x=-2[\/latex], [latex](-2)^2 \\leq 3(-2)+4 \\rightarrow 4 \\leq -2[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a\u00a0solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=0[\/latex], [latex](0)^2 \\leq 3(0)+4 \\rightarrow 0 \\leq 4[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=5[\/latex], [latex](5)^2 \\leq 3(5)+4 \\rightarrow 25 \\leq 19[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\r\n<img class=\"aligncenter wp-image-1160\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1.png\" alt=\"Number line plotted -1, 4 (closed) as solutions, marked -2, 0, 5 as test values, middle interval is marked in red\" width=\"391\" height=\"78\" \/>\r\n<p style=\"padding-left: 30px;\">We can see the results of test values are true from the graph of [latex]f(x)=x^2-3x-4[\/latex] below:<\/p>\r\n<img class=\"aligncenter wp-image-1161\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-2.png\" alt=\"Graph of f(x)=x^2-3x-4, marked below x-axis in red\" width=\"194\" height=\"246\" \/>\r\n\r\nTherefore, the solution of the polynomial inequality\u00a0[latex]x^2 \\leq 3x+4[\/latex] is [latex][-1, 4][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: No solution Case<\/h3>\r\nSolve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.\r\n<p style=\"padding-left: 30px;\">[latex]x^6 +2 &lt; 0[\/latex]<\/p>\r\n[reveal-answer q=\"571875\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"571875\"]\r\n\r\nFirst, let's rewrite the inequality as an equation and then solve it:\r\n<p style=\"padding-left: 30px;\">[latex]x^6 + 2 = 0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^6 \\ne -2[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">So, there is no solution for this polynomial equation.<\/p>\r\nSecond, since there is no solution, we cannot plot anything on the number line, and there is only one interval on it: [latex](-\\infty, \\infty)\r\n\r\n<img class=\"aligncenter wp-image-1162\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png\" alt=\"Number line no solution plotted, marked 0 as test value\" width=\"307\" height=\"64\" \/>\r\n\r\nThird, choose your choice of test value from the interval:\r\n<p style=\"padding-left: 30px;\">I chose [latex] 0[\/latex].<\/p>\r\nFourth, test the interval using the test value:\r\n<p style=\"padding-left: 30px;\">When [latex]x=0[\/latex], [latex](0)^6+2&lt;0 \\rightarrow 2&lt;0[\/latex] <span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\r\n<img class=\"aligncenter wp-image-1162\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png\" alt=\"Number line no solution plotted, marked 0 as test value\" width=\"307\" height=\"64\" \/>\r\n<p style=\"padding-left: 30px;\">We can see the result of test value is true from the graph of [latex]f(x)=x^2-3x-4[\/latex] below. The graph is above the [latex]x[\/latex]-axis everywhere:<\/p>\r\n<img class=\"aligncenter wp-image-1163\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-1.png\" alt=\"Graph of f(x)=x^6+2, marked below x-axis in red\" width=\"216\" height=\"166\" \/>\r\n\r\nTherefore, the solution of the polynomial inequality\u00a0[latex]x^6+2&gt;0[\/latex] is <strong>no solution<\/strong>.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSolve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.\r\n<p style=\"padding-left: 30px;\">[latex]-\\frac{1}{3}x^2 \\leq 0[\/latex]<\/p>\r\n[reveal-answer q=\"182644\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"182644\"]\r\n\r\nFirst, let's rewrite the inequality as an equation and then solve it:\r\n<p style=\"padding-left: 30px;\">[latex]-\\frac{1}{3}x^2 = 0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x^2=0[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]x=0[\/latex]<\/p>\r\nSecond,\u00a0plot the solution on the number line. Since the inequality has \"[latex]\\geq[\/latex]\" (greater than or equal to), which has \"[latex]=[\/latex]\" (equal sign), it should be <strong>closed<\/strong> circles:\r\n\r\n<img class=\"aligncenter wp-image-1164\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4.png\" alt=\"Number line plotted 0 (closed) as solution, marked -1, 1 as test values,\" width=\"325\" height=\"77\" \/>\r\n\r\nThird, choose your choice of test value from the interval:\r\n<p style=\"padding-left: 30px;\">I chose [latex] -1[\/latex] and [latex]1[\/latex].<\/p>\r\nFourth, test the interval using the test value:\r\n<p style=\"padding-left: 30px;\">When [latex]x=-1[\/latex], [latex]-\\frac{1}{3}(-1)^2 \\leq 0 \\rightarrow -\\frac{1}{3} \\leq 0[\/latex] <span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<p style=\"padding-left: 30px;\">When [latex]x=1[\/latex], [latex]-\\frac{1}{3}(1)^2 \\leq 0 \\rightarrow -\\frac{1}{3} \\leq 0[\/latex] <span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\r\n<img class=\"aligncenter wp-image-1165\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1.png\" alt=\"Number line plotted 0 (closed) as solution, marked -1, 1 as test values, whole number line is marked in red\" width=\"324\" height=\"83\" \/>\r\n<p style=\"padding-left: 30px;\">We can see the result of test values are true from the graph of [latex]f(x)=-\\frac{1}{3} x^2[\/latex] below:<\/p>\r\n<img class=\"aligncenter wp-image-1166\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2.png\" alt=\"Graph of f(x)=-1\/3x^2, marked below x-axis in red\" width=\"235\" height=\"184\" \/>\r\n<p style=\"padding-left: 30px;\">*\u00a0The graph is below the [latex]x[\/latex]-axis everywhere except at [latex]x=0[\/latex]. However, since\u00a0the inequality has \"[latex]\\leq[\/latex],\" which has \"[latex]=[\/latex]\" (equal sign), [latex]x=0[\/latex] also satisfies the inequality.<\/p>\r\nTherefore, the solution of the polynomial inequality\u00a0[latex]-\\frac{1}{3} x^2[\/latex] is [latex](-\\infty, \\infty)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Solve polynomial inequalities using boundary value method.<\/li>\n<\/ul>\n<\/div>\n<h2>Solving Polynomial Inequalities using Boundary Value Method<\/h2>\n<p>Any inequality that can be put into one of the following forms<\/p>\n<p style=\"padding-left: 30px;\">[latex]f(x)>0, f(x) \\geq 0, f(x)<0,[\/latex] or [latex]f(x) \\leq 0[\/latex], where [latex]f[\/latex] is a polynomial function<\/p>\n<p>is called <strong>polynomial inequality<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>How To: Solve Polynomial Inequalities using Boundary Value MEthod<\/h3>\n<ol>\n<li>Rewrite the given polynomial inequality as an equation by replacing the inequality symbol with the equal sign.<\/li>\n<li>Solve the polynomial equation. The real solution(s) of the equation is(are) the <strong>boundary point(s)<\/strong>.<\/li>\n<li>Plot the boundary point(s) from Step 2 on a number line.<br \/>\n[latex]\\Rightarrow[\/latex] Use an open circle when the given inequality has [latex]<[\/latex] or [latex]>[\/latex]<br \/>\n[latex]\\Rightarrow[\/latex] Use a closed circle when the given inequality has [latex]\\leq[\/latex] or [latex]\\geq[\/latex].<\/li>\n<li>Choose one number, which is called a <strong>test value<\/strong>, from each interval and test the intervals by evaluating the given inequality at that number.<br \/>\n[latex]\\Rightarrow[\/latex] If the inequality is TRUE, then the interval is a solution of the inequality.<br \/>\n[latex]\\Rightarrow[\/latex] If the inequality is FALSE, then the interval is not a solution of the inequality.<\/li>\n<li>Write the solution set (usually in interval notation), selecting the interval(s) from Step 4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving Polynomial Inequality using Boundary Value Method<\/h3>\n<p>Solve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^4-4x^3+3x^2 > 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q933923\">Show Solution<\/span><\/p>\n<div id=\"q933923\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, let&#8217;s rewrite the inequality as an equation and then solve it:<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^4-4x^3+3x^2=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2(x^2-4x+3)=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2(x-1)(x-3)=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2=0, x-1=0[\/latex] or [latex]x-3=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x=0, x=1[\/latex] or [latex]x=3[\/latex]<\/p>\n<p>Second, plot the solutions on the number line. Since the inequality has &#8220;[latex]>[\/latex]&#8221; (greater than), which doesn&#8217;t have &#8220;[latex]=[\/latex]&#8221; (equal sign), they should be <strong>open<\/strong> circles:<\/p>\n<p style=\"padding-left: 30px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1154\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1.png\" alt=\"Numberline plotted 0, 1, 3 (open) as solutions, marked -1, 0.5, 2, 5 as testing values\" width=\"453\" height=\"74\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1.png 606w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-300x49.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-65x11.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-225x37.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-350x57.png 350w\" sizes=\"auto, (max-width: 453px) 100vw, 453px\" \/><\/p>\n<p style=\"padding-left: 30px;\">Note that now there are four intervals on the number line: [latex](-\\infty, 0), (0, 1), (1, 3),[\/latex] and [latex](3, \\infty)[\/latex]<\/p>\n<p>Third, choose your choice of test value from each interval:<\/p>\n<p style=\"padding-left: 30px;\">I chose [latex]-1, 0.5, 2,[\/latex] and [latex]5[\/latex] from those four intervals.<\/p>\n<p>Fourth, test each interval using the test value:<\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=-1[\/latex], [latex](-1)^4-4(-1)^3+3(-1)^2>0 \\rightarrow 8>0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=0.5[\/latex], [latex](0.5)^4-4(0.5)^3+3(0.5)^2>0 \\rightarrow 0.3125>0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=2[\/latex], [latex](2)^4-4(2)^3+3(2)^2>0 \\rightarrow -4>0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=5[\/latex], [latex](5)^4-4(5)^3+3(5)^2>0 \\rightarrow 200>0[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1157\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2.png\" alt=\"Number line plotted 0, 1, 3 (open) as solutions, marked -1, 0.5, 2, 5 as testing values, first and fourth intervals are marked in red\" width=\"457\" height=\"71\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2.png 605w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2-300x47.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2-65x10.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2-225x35.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-2-350x54.png 350w\" sizes=\"auto, (max-width: 457px) 100vw, 457px\" \/><\/p>\n<p style=\"padding-left: 30px;\">Actually, we can check the results of test values from the graph of [latex]f(x)=x^4-4x^3+3x^2[\/latex] below:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1158\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3.png\" alt=\"Graph of f(x)=x^4-4x^3+3x^2, marked above x-axis in red\" width=\"237\" height=\"284\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3.png 286w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3-250x300.png 250w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3-65x78.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-1-3-225x270.png 225w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/p>\n<p style=\"padding-left: 30px;\">* As we can see from the graph, it doesn&#8217;t matter which number you choose from each interval for testing. For example, any number that is greater than 5 will make the function value positive.<\/p>\n<p>Therefore, the solution of the polynomial inequality\u00a0[latex]x^4-4x^3+3x^2 > 0[\/latex] is [latex](-\\infty, 0) \\cup (0, 1) \\cup (3, \\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2 \\leq 3x+4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q651999\">Show Solution<\/span><\/p>\n<div id=\"q651999\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, let&#8217;s rewrite the inequality as an equation and then solve it:<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2 = 3x+4[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2-3x-4=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex](x+1)(x-4)=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x+1=0[\/latex] or [latex]x-4=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x=-1[\/latex] or [latex]x=4[\/latex]<\/p>\n<p>Second, plot the solutions on the number line. Since the inequality has &#8220;[latex]\\leq[\/latex]&#8221; (less than or equal to), which has &#8220;[latex]=[\/latex]&#8221; (equal sign), they should be <strong>closed<\/strong> circles:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1159\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2.png\" alt=\"Number line plotted -1, 4 (closed) as solutions, marked -2, 0, 5 as test values\" width=\"392\" height=\"76\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2.png 562w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-300x58.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-65x13.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-225x44.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-350x68.png 350w\" sizes=\"auto, (max-width: 392px) 100vw, 392px\" \/><\/p>\n<p style=\"padding-left: 30px;\">Note that now there are three intervals on the number line: [latex](-\\infty, -1], [-1, 4],[\/latex] and [latex][4, \\infty)[\/latex]<\/p>\n<p>Third, choose your choice of test value from each interval:<\/p>\n<p style=\"padding-left: 30px;\">I chose [latex]-2, 0,[\/latex] and [latex]5[\/latex] from those three intervals.<\/p>\n<p>Fourth, test each interval using the test value:<\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=-2[\/latex], [latex](-2)^2 \\leq 3(-2)+4 \\rightarrow 4 \\leq -2[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a\u00a0solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=0[\/latex], [latex](0)^2 \\leq 3(0)+4 \\rightarrow 0 \\leq 4[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=5[\/latex], [latex](5)^2 \\leq 3(5)+4 \\rightarrow 25 \\leq 19[\/latex]\u00a0<span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1160\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1.png\" alt=\"Number line plotted -1, 4 (closed) as solutions, marked -2, 0, 5 as test values, middle interval is marked in red\" width=\"391\" height=\"78\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1.png 577w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1-300x60.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1-65x13.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1-225x45.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-1-350x70.png 350w\" sizes=\"auto, (max-width: 391px) 100vw, 391px\" \/><\/p>\n<p style=\"padding-left: 30px;\">We can see the results of test values are true from the graph of [latex]f(x)=x^2-3x-4[\/latex] below:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1161\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-2.png\" alt=\"Graph of f(x)=x^2-3x-4, marked below x-axis in red\" width=\"194\" height=\"246\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-2.png 234w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-2-65x82.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-2-2-225x285.png 225w\" sizes=\"auto, (max-width: 194px) 100vw, 194px\" \/><\/p>\n<p>Therefore, the solution of the polynomial inequality\u00a0[latex]x^2 \\leq 3x+4[\/latex] is [latex][-1, 4][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: No solution Case<\/h3>\n<p>Solve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^6 +2 < 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q571875\">Show Solution<\/span><\/p>\n<div id=\"q571875\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, let&#8217;s rewrite the inequality as an equation and then solve it:<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^6 + 2 = 0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^6 \\ne -2[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">So, there is no solution for this polynomial equation.<\/p>\n<p>Second, since there is no solution, we cannot plot anything on the number line, and there is only one interval on it: [latex](-\\infty, \\infty)    <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1162\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png\" alt=\"Number line no solution plotted, marked 0 as test value\" width=\"307\" height=\"64\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png 407w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-300x63.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-65x14.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-225x47.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-350x73.png 350w\" sizes=\"auto, (max-width: 307px) 100vw, 307px\" \/>    Third, choose your choice of test value from the interval:  <\/p>\n<p style=\"padding-left: 30px;\">I chose [latex] 0[\/latex].<\/p>\n<p>Fourth, test the interval using the test value:<\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=0[\/latex], [latex](0)^6+2<0 \\rightarrow 2<0[\/latex] <span style=\"color: #ff0000;\"><strong>(False)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>not a solution<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1162\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png\" alt=\"Number line no solution plotted, marked 0 as test value\" width=\"307\" height=\"64\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3.png 407w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-300x63.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-65x14.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-225x47.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-350x73.png 350w\" sizes=\"auto, (max-width: 307px) 100vw, 307px\" \/><\/p>\n<p style=\"padding-left: 30px;\">We can see the result of test value is true from the graph of [latex]f(x)=x^2-3x-4[\/latex] below. The graph is above the [latex]x[\/latex]-axis everywhere:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1163\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-1.png\" alt=\"Graph of f(x)=x^6+2, marked below x-axis in red\" width=\"216\" height=\"166\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-1.png 279w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-1-65x50.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-3-1-225x173.png 225w\" sizes=\"auto, (max-width: 216px) 100vw, 216px\" \/><\/p>\n<p>Therefore, the solution of the polynomial inequality\u00a0[latex]x^6+2>0[\/latex] is <strong>no solution<\/strong>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Solve the polynomial inequality using boundary value method. Graph the solution set and write the solution in interval notation.<\/p>\n<p style=\"padding-left: 30px;\">[latex]-\\frac{1}{3}x^2 \\leq 0[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182644\">Show Solution<\/span><\/p>\n<div id=\"q182644\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, let's rewrite the inequality as an equation and then solve it:<\/p>\n<p style=\"padding-left: 30px;\">[latex]-\\frac{1}{3}x^2 = 0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x^2=0[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">[latex]x=0[\/latex]<\/p>\n<p>Second,\u00a0plot the solution on the number line. Since the inequality has \"[latex]\\geq[\/latex]\" (greater than or equal to), which has \"[latex]=[\/latex]\" (equal sign), it should be <strong>closed<\/strong> circles:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1164\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4.png\" alt=\"Number line plotted 0 (closed) as solution, marked -1, 1 as test values,\" width=\"325\" height=\"77\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4.png 451w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-300x71.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-65x15.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-225x53.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-350x83.png 350w\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" \/><\/p>\n<p>Third, choose your choice of test value from the interval:<\/p>\n<p style=\"padding-left: 30px;\">I chose [latex]-1[\/latex] and [latex]1[\/latex].<\/p>\n<p>Fourth, test the interval using the test value:<\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=-1[\/latex], [latex]-\\frac{1}{3}(-1)^2 \\leq 0 \\rightarrow -\\frac{1}{3} \\leq 0[\/latex] <span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p style=\"padding-left: 30px;\">When [latex]x=1[\/latex], [latex]-\\frac{1}{3}(1)^2 \\leq 0 \\rightarrow -\\frac{1}{3} \\leq 0[\/latex] <span style=\"color: #ff0000;\"><strong>(True)<\/strong><\/span>\u00a0 [latex]\\therefore[\/latex] <strong>solution<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1165\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1.png\" alt=\"Number line plotted 0 (closed) as solution, marked -1, 1 as test values, whole number line is marked in red\" width=\"324\" height=\"83\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1.png 452w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1-300x77.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1-65x17.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1-225x58.png 225w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-1-350x90.png 350w\" sizes=\"auto, (max-width: 324px) 100vw, 324px\" \/><\/p>\n<p style=\"padding-left: 30px;\">We can see the result of test values are true from the graph of [latex]f(x)=-\\frac{1}{3} x^2[\/latex] below:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1166\" src=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2.png\" alt=\"Graph of f(x)=-1\/3x^2, marked below x-axis in red\" width=\"235\" height=\"184\" srcset=\"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2.png 326w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2-300x235.png 300w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2-65x51.png 65w, https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-content\/uploads\/sites\/5858\/2023\/07\/2.6-Poly-Ineq-4-2-225x176.png 225w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/p>\n<p style=\"padding-left: 30px;\">*\u00a0The graph is below the [latex]x[\/latex]-axis everywhere except at [latex]x=0[\/latex]. However, since\u00a0the inequality has \"[latex]\\leq[\/latex],\" which has \"[latex]=[\/latex]\" (equal sign), [latex]x=0[\/latex] also satisfies the inequality.<\/p>\n<p>Therefore, the solution of the polynomial inequality\u00a0[latex]-\\frac{1}{3} x^2[\/latex] is [latex](-\\infty, \\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1145\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li> Solving Polynomial Inequalities. <strong>Authored by<\/strong>: Michelle Eunhee Chung. <strong>Provided by<\/strong>: Georgia State University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":705214,"menu_order":36,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\" Solving Polynomial Inequalities\",\"author\":\"Michelle Eunhee Chung\",\"organization\":\"Georgia State University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mchung12"],"pb_section_license":""},"chapter-type":[],"contributor":[62],"license":[],"class_list":["post-1145","chapter","type-chapter","status-publish","hentry","contributor-mchung12"],"part":91,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1145","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/705214"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1145\/revisions"}],"predecessor-version":[{"id":1406,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1145\/revisions\/1406"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/91"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1145\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=1145"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1145"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1145"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=1145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}