document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200712>Question 1200712</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the answer to thus quadratic inequality. 3(x^2-1)>-8x. The solution should be written in interval notation.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834915 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=834915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>(-∞, -3) U (1/3, ∞)</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Let's get everything to one side<BR>\n" );
document.write( "3(x^2-1) > -8x<BR>\n" );
document.write( "3x^2-3 > -8x<BR>\n" );
document.write( "3x^2-3 + 8x > 0<BR>\n" );
document.write( "3x^2 + 8x - 3 > 0\r<BR>\n" );
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document.write( "To solve that inequality, we'll consider the corresponding equation<BR>\n" );
document.write( "3x^2 + 8x - 3 = 0<BR>\n" );
document.write( "We need to find the roots or x intercepts.\r<BR>\n" );
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document.write( "Factoring may or may not be possible. <BR>\n" );
document.write( "The trial-and-error factoring approach is something I'm not fond of, so I prefer the quadratic formula instead.<BR>\n" );
document.write( "Plug in:<BR>\n" );
document.write( "a = 3<BR>\n" );
document.write( "b = 8<BR>\n" );
document.write( "c = -3<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-8%2B10%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%28-8%2B10%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-8-10%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%28-8-10%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-18%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%28-18%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F3 ALIGN=MIDDLE  ALT=\"x+=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3 ALIGN=MIDDLE  ALT=\"x+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Because each root is rational, it turns out that we could have factored previously.<BR>\n" );
document.write( "The x = 1/3 leads to 3x = 1 and further leads to 3x-1 = 0<BR>\n" );
document.write( "The x = -3 leads to x+3 = 0<BR>\n" );
document.write( "We have the factors (3x-1) and (x+3)<BR>\n" );
document.write( "Therefore, 3x^2 + 8x - 3 = (3x-1)(x+3)<BR>\n" );
document.write( "You can use the FOIL rule on (3x-1)(x+3) to get 3x^2 + 8x - 3 again.\r<BR>\n" );
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document.write( "Anyways, the roots we found were:<BR>\n" );
document.write( "x = -3<BR>\n" );
document.write( "x = 1/3\r<BR>\n" );
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document.write( "Draw out a number line. <BR>\n" );
document.write( "Mark -3 on it, and also mark 1/3<BR>\n" );
document.write( "We'll label 3 regions<BR>\n" );
document.write( "Region A = stuff to the left of -3<BR>\n" );
document.write( "Region B = stuff between the number -3 and the number 1/3<BR>\n" );
document.write( "Region C = stuff to the right of 1/3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28600%2C+200%2C+-7%2C4%2C-5%2C5%2C%0D%0Aline%28-6-3%2C0%2C3%2B3%2C0%29%2C%0D%0Aline%28-6%2C0.2%2C-6%2C-0.2%29%2C%0D%0Aline%28-5%2C0.2%2C-5%2C-0.2%29%2C%0D%0Aline%28-4%2C0.2%2C-4%2C-0.2%29%2C%0D%0Aline%28-3%2C0.2%2C-3%2C-0.2%29%2C%0D%0Aline%28-2%2C0.2%2C-2%2C-0.2%29%2C%0D%0Aline%28-1%2C0.2%2C-1%2C-0.2%29%2C%0D%0Aline%280%2C0.2%2C0%2C-0.2%29%2C%0D%0Aline%281%2C0.2%2C1%2C-0.2%29%2C%0D%0Aline%282%2C0.2%2C2%2C-0.2%29%2C%0D%0Aline%283%2C0.2%2C3%2C-0.2%29%2C%0D%0Alocate%28-6.2%2C-0.2%2C%22-6%22%29%2C%0D%0Alocate%28-5.2%2C-0.2%2C%22-5%22%29%2C%0D%0Alocate%28-4.2%2C-0.2%2C%22-4%22%29%2C%0D%0Alocate%28-3.2%2C-0.2%2C%22-3%22%29%2C%0D%0Alocate%28-2.2%2C-0.2%2C%22-2%22%29%2C%0D%0Alocate%28-1.2%2C-0.2%2C%22-1%22%29%2C%0D%0Alocate%28-0.2%2B0.092%2C-0.2-0.05%2C%220%22%29%2C%0D%0Alocate%280.8%2B0.092%2C-0.2-0.05%2C%221%22%29%2C%0D%0Alocate%281.8%2B0.092%2C-0.2-0.05%2C%222%22%29%2C%0D%0Alocate%282.8%2B0.092%2C-0.2-0.05%2C%223%22%29%2C%0D%0A%0D%0Ared%28circle%28-3%2C0%2C0.04%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.06%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.08%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.10%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.12%29%29%2C%0D%0A%0D%0Ared%28circle%281%2F3%2C0%2C0.04%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.06%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.08%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.10%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.12%29%29%2C%0D%0A%0D%0Ared%28locate%28-3%2C0%2B1.05%2C%22-3%22%29%29%2C%0D%0Ared%28locate%281%2F3%2C0%2B1.05%2C%221%2F3%22%29%29%2C%0D%0A%0D%0Ablue%28locate%28-5%2C0%2B1.25%2C%22A%22%29%29%2C%0D%0Ablue%28locate%28%28-3%2B1%2F3%29%2F2%2C0%2B1.25%2C%22B%22%29%29%2C%0D%0Ablue%28locate%282%2C0%2B1.25%2C%22C%22%29%29%0D%0A%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Pick a value from region A to test.<BR>\n" );
document.write( "I'll go for x = -4<BR>\n" );
document.write( "3(x^2-1) > -8x<BR>\n" );
document.write( "3((-4)^2-1) > -8(-4)<BR>\n" );
document.write( "3(16-1) > 32<BR>\n" );
document.write( "3(15) > 32<BR>\n" );
document.write( "45 > 32<BR>\n" );
document.write( "The final result is a true statement.<BR>\n" );
document.write( "Therefore, 3(x^2-1)>-8x is true when x = -4<BR>\n" );
document.write( "Furthermore, 3(x^2-1)>-8x is true for any value in region A.<BR>\n" );
document.write( "We can write x < -3 as part of the solution set\r<BR>\n" );
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document.write( "Now let's test region B.<BR>\n" );
document.write( "I'll use x = 0<BR>\n" );
document.write( "3(x^2-1) > -8x<BR>\n" );
document.write( "3(0^2-1) > -8*0<BR>\n" );
document.write( "3(0-1) > 0<BR>\n" );
document.write( "3(-1) > 0<BR>\n" );
document.write( "-3 > 0<BR>\n" );
document.write( "That is false, so region B is crossed off the list.\r<BR>\n" );
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document.write( "Lastly we need to test region C.<BR>\n" );
document.write( "I'll pick x = 2.<BR>\n" );
document.write( "3(x^2-1) > -8x<BR>\n" );
document.write( "3(2^2-1) > -8*2<BR>\n" );
document.write( "3(4-1) > -16<BR>\n" );
document.write( "3(3) > -16<BR>\n" );
document.write( "9 > -16<BR>\n" );
document.write( "This is true, which makes x > 1/3 also part of the solution set.\r<BR>\n" );
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document.write( "Our solution set consists of x values such that<BR>\n" );
document.write( "x < -3 or x > 1/3\r<BR>\n" );
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document.write( "We can rewrite x < -3 as -∞ < x < -3<BR>\n" );
document.write( "We can rewrite x > 1/3 as 1/3 < x < ∞<BR>\n" );
document.write( "Both of these help us get toward interval notation.\r<BR>\n" );
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document.write( "-∞ < x < -3 in interval notation is (-∞, -3)<BR>\n" );
document.write( "1/3 < x < ∞ in interval notation is (1/3, ∞)\r<BR>\n" );
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document.write( "Those disjoint intervals are glued together with the union operator to arrive at the <font color=red>final answer of (-∞, -3) U (1/3, ∞)</font>\r<BR>\n" );
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document.write( "This is what the solution set looks like when graphed on a number line:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28600%2C+200%2C+-7%2C4%2C-5%2C5%2C%0D%0Aline%28-6-3%2C0%2C3%2B3%2C0%29%2C%0D%0Aline%28-6%2C0.2%2C-6%2C-0.2%29%2C%0D%0Aline%28-5%2C0.2%2C-5%2C-0.2%29%2C%0D%0Aline%28-4%2C0.2%2C-4%2C-0.2%29%2C%0D%0A%0D%0Aline%28-2%2C0.2%2C-2%2C-0.2%29%2C%0D%0Aline%28-1%2C0.2%2C-1%2C-0.2%29%2C%0D%0Aline%280%2C0.2%2C0%2C-0.2%29%2C%0D%0Aline%281%2C0.2%2C1%2C-0.2%29%2C%0D%0Aline%282%2C0.2%2C2%2C-0.2%29%2C%0D%0Aline%283%2C0.2%2C3%2C-0.2%29%2C%0D%0Alocate%28-6.2%2C-0.2-0.05%2C%22-6%22%29%2C%0D%0Alocate%28-5.2%2C-0.2-0.05%2C%22-5%22%29%2C%0D%0Alocate%28-4.2%2C-0.2-0.05%2C%22-4%22%29%2C%0D%0Alocate%28-3.2%2C-0.2-0.05%2C%22-3%22%29%2C%0D%0Alocate%28-2.2%2C-0.2-0.05%2C%22-2%22%29%2C%0D%0Alocate%28-1.2%2C-0.2-0.05%2C%22-1%22%29%2C%0D%0Alocate%28-0.2%2B0.092%2C-0.2-0.05%2C%220%22%29%2C%0D%0Alocate%280.8%2B0.092%2C-0.2-0.05%2C%221%22%29%2C%0D%0Alocate%281.8%2B0.092%2C-0.2-0.05%2C%222%22%29%2C%0D%0Alocate%282.8%2B0.092%2C-0.2-0.05%2C%223%22%29%2C%0D%0A%0D%0Ared%28circle%28-3%2C0%2C0.10%29%29%2C%0D%0Ared%28circle%28-3%2C0%2C0.12%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.10%29%29%2C%0D%0Ared%28circle%281%2F3%2C0%2C0.12%29%29%2C%0D%0A%0D%0Alocate%281%2F3%2C0%2B1.05%2C%221%2F3%22%29%2C%0D%0A%0D%0Ared%28locate%28-5%2C0%2B1.25%2C%22x+%3C+-3%22%29%29%2C%0D%0Ared%28locate%282%2C0%2B1.25%2C%22x%3E1%2F3%22%29%29%2C%0D%0A%0D%0A%0D%0Ared%28line%281%2F3%2B0.12%2C-0.1%2C6%2C-0.1%29%29%2C%0D%0Ared%28line%281%2F3%2B0.12%2C-0.05%2C6%2C-0.05%29%29%2C%0D%0Ared%28line%281%2F3%2B0.12%2C0%2C6%2C0%29%29%2C%0D%0Ared%28line%281%2F3%2B0.12%2C0.05%2C6%2C0.05%29%29%2C%0D%0Ared%28line%281%2F3%2B0.12%2C0.1%2C6%2C0.1%29%29%2C%0D%0A%0D%0Ared%28line%284.25-0%2C0%2C3.5-0%2C1%29%29%2C%0D%0Ared%28line%284.25-0.02%2C0%2C3.5-0.02%2C1%29%29%2C%0D%0Ared%28line%284.25-0.04%2C0%2C3.5-0.04%2C1%29%29%2C%0D%0Ared%28line%284.25-0.06%2C0%2C3.5-0.06%2C1%29%29%2C%0D%0Ared%28line%284.25-0.08%2C0%2C3.5-0.08%2C1%29%29%2C%0D%0Ared%28line%284.25-0.1%2C0%2C3.5-0.1%2C1%29%29%2C%0D%0A%0D%0Ared%28line%284.25-0%2C0%2C3.5-0%2C-1%29%29%2C%0D%0Ared%28line%284.25-0.02%2C0%2C3.5-0.02%2C-1%29%29%2C%0D%0Ared%28line%284.25-0.04%2C0%2C3.5-0.04%2C-1%29%29%2C%0D%0Ared%28line%284.25-0.06%2C0%2C3.5-0.06%2C-1%29%29%2C%0D%0Ared%28line%284.25-0.08%2C0%2C3.5-0.08%2C-1%29%29%2C%0D%0Ared%28line%284.25-0.1%2C0%2C3.5-0.1%2C-1%29%29%2C%0D%0A%0D%0A%0D%0Ared%28line%28-3-0.12%2C-0.1%2C-7%2C-0.1%29%29%2C%0D%0Ared%28line%28-3-0.12%2C-0.05%2C-7%2C-0.05%29%29%2C%0D%0Ared%28line%28-3-0.12%2C0%2C-7%2C0%29%29%2C%0D%0Ared%28line%28-3-0.12%2C0.05%2C-7%2C0.05%29%29%2C%0D%0Ared%28line%28-3-0.12%2C0.1%2C-7%2C0.1%29%29%2C%0D%0A%0D%0Ared%28line%28-7%2B0%2C0%2C-6.25%2B0%2C1%29%29%2C%0D%0Ared%28line%28-7%2B0.02%2C0%2C-6.25%2B0.02%2C1%29%29%2C%0D%0Ared%28line%28-7%2B0.04%2C0%2C-6.25%2B0.04%2C1%29%29%2C%0D%0Ared%28line%28-7%2B0.06%2C0%2C-6.25%2B0.06%2C1%29%29%2C%0D%0Ared%28line%28-7%2B0.08%2C0%2C-6.25%2B0.08%2C1%29%29%2C%0D%0Ared%28line%28-7%2B0.1%2C0%2C-6.25%2B0.1%2C1%29%29%2C%0D%0A%0D%0Ared%28line%28-7%2B0%2C0%2C-6.25%2B0%2C-1%29%29%2C%0D%0Ared%28line%28-7%2B0.02%2C0%2C-6.25%2B0.02%2C-1%29%29%2C%0D%0Ared%28line%28-7%2B0.04%2C0%2C-6.25%2B0.04%2C-1%29%29%2C%0D%0Ared%28line%28-7%2B0.06%2C0%2C-6.25%2B0.06%2C-1%29%29%2C%0D%0Ared%28line%28-7%2B0.08%2C0%2C-6.25%2B0.08%2C-1%29%29%2C%0D%0Ared%28line%28-7%2B0.1%2C0%2C-6.25%2B0.1%2C-1%29%29%2C%0D%0A%0D%0Ared%28locate%28-5.5%2C3%2Cmatrix%281%2C5%2C%22%28%22%2C-infinity%2C%22%2C%22%2C-3%2C%22%29%22%29+%29%29%2C%0D%0Ared%28locate%28-5%2C2%2C%22aka%22%29+%29%2C%0D%0A%0D%0Ared%28locate%281.5%2C3%2Cmatrix%281%2C5%2C%22%28%22%2C%221%2F3%22%2C%22%2C%22%2Cinfinity%2C%22%29%22%29+%29%29%2C%0D%0Ared%28locate%282%2C2%2C%22aka%22%29+%29%0D%0A%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Take note of the open holes at -3 and at 1/3.<BR>\n" );
document.write( "Verbally we can describe the graph as having \"open holes at -3 and at 1/3, with shading everywhere but between those open holes\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "An alternative graph to plot is y = 3x^2+8x-3<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C3%2C-10%2C5%2C%0D%0Agraph%28400%2C400%2C-6%2C3%2C-10%2C5%2C-1000%2C3x%5E2+%2B+8x+-+3%29%2C%0D%0Acircle%28-3%2C0%2C0.06%29%2C%0D%0Acircle%28-3%2C0%2C0.08%29%2C%0D%0Acircle%28-3%2C0%2C0.10%29%2C%0D%0Acircle%28-3%2C0%2C0.12%29%2C%0D%0Acircle%28-3%2C0%2C0.14%29%2C%0D%0A%0D%0Acircle%281%2F3%2C0%2C0.06%29%2C%0D%0Acircle%281%2F3%2C0%2C0.08%29%2C%0D%0Acircle%281%2F3%2C0%2C0.10%29%2C%0D%0Acircle%281%2F3%2C0%2C0.12%29%2C%0D%0Acircle%281%2F3%2C0%2C0.14%29%2C%0D%0A%0D%0Alocate%28-3%2C0%2B0.75%2C%22%28-3%2C0%29%22%29%2C%0D%0Alocate%281%2F3%2C0%2B0.75%2C%22%281%2F3%2C0%29%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The parabola is above the x axis when either x < -3 or when x > 1/3.<BR>\n" );
document.write( "In other words, the parabola is on or below the x axis when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+%3C=+x+%3C=+1%2F3 ALIGN=MIDDLE  ALT=\"-3+%3C=+x+%3C=+1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Otherwise, it is above the x axis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Desmos and GeoGebra are two graphing options I recommend.<BR>\n" );
document.write( "The graphing option allows a person to quickly arrive at the solution set.<BR>\n" );
document.write( "However, I recommend following an algebraic approach and then use a graph to verify (rather than solely rely on a graph to do all the work for you).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A similar problem is found <a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/playground/test.faq.question.1200141.html\">here</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "See <a  rel=nofollow HREF= \"https://www.mathsisfun.com/algebra/inequality-quadratic-solving.html\">this article</a> for further reading.<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );